Quantumplations

Problem #\(1\): Consider a simplistic classical model of an atom as a positively charged nucleus \(Q>0\) surrounded by a spherical, uniformly dense electron cloud \(-Q<0\) of radius \(a\). If this atom is subjected to a DC external electric field \(\textbf E_{\text{ext}}\), show that the induced dipole \(\textbf p_{\text{ind}}\) developed by the atom is given by \(\textbf p_{\text{ind}}=\alpha\textbf E_{\text{ext}}\) where the atomic polarizability is given by \(\alpha=4\pi\varepsilon_0 a^3>0\). Neglect any higher-order multipole moments of the electron cloud (i.e. assume it maintains a spherical shape when perturbed by \(\textbf E_{\text{ext}}\)).

Solution #\(1\): By Gauss’s law for electric fields, the internal electrostatic field \(\textbf E_{\text{int}}(\textbf x)\) due to just the electron cloud must rise linearly in magnitude to match the Coulomb field at \(|\textbf x|=a\), thus \(\textbf E_{\text{int}}(|\textbf x|=a)=-Q/4\pi\varepsilon_0 a^2\hat{\textbf x}\), hence \(\textbf E_{\text{int}}(\textbf x)=-Q|\textbf x|/4\pi\varepsilon_0 a^3\hat{\textbf x}\) for \(|\textbf x|\leq a\). When the external electric field \(\textbf E_{\text{ext}}\) is applied, this will induce a displacement of the nucleus to a new equilibrium \(\textbf x=\textbf 0\mapsto\textbf x=\Delta\textbf x_{\text{ind}}\) relative to the center of the electron cloud, at which location force balance requires:

\[\textbf E_{\text{int}}(\Delta\textbf x_{\text{ind}})+\textbf E_{\text{ext}}=\textbf 0\]

Noting that the induced dipole is \(\textbf p_{\text{ind}}=Q\Delta\textbf x_{\text{ind}}\), the claim follows.

Problem #\(2\): Explain what it means for a dielectric to be linear and explain under what circumstances can most dielectrics be modelled as linear.

Solution #\(2\): A dielectric is said to be linear iff, when subjected to an external (not necessarily uniform or DC!) electric field \(\textbf E_{\text{ext}}\), the induced polarization \(\textbf P_{\text{ind}}\) is linear in \(\textbf E_{\text{ext}}\) in the sense that there exists an electric susceptibility \(\chi_e\) such that:

\[\textbf P_{\text{ind}}=\varepsilon_0\chi_e\textbf E_{\text{ext}}\]

By definition, \(\textbf P_{\text{ind}}:=n_{\textbf p_{\text{ind}}}\textbf p_{\text{ind}}\) is the number density \(n_{\textbf p_{\text{ind}}}\) of induced electric dipoles \(\textbf p_{\text{ind}}\). If \(\textbf E_{\text{ext}}\) is DC as in Problem #\(1\), then in that case \(\chi_e>0\) is real and given by \(\chi_e=4\pi n_{\textbf p_{\text{ind}}}a^3\). In general, the assumption of linearity is only a good approximation provided \(|\textbf E_{\text{ext}}|\) is not too strong.

Problem #\(3\): Explain what the concept of bound charge is and show that the surface bound charge density \(\sigma_b\) and the volume bound charge density \(\rho_b\) are related to the polarization \(\textbf P\) via:

\[\sigma_b=\hat{\textbf n}\cdot\textbf P\]

\[\rho_b=-\frac{\partial}{\partial\textbf x}\cdot\textbf P\]

Comment on the case where \(\textbf P\) is uniform within a material.

Solution #\(3\): Whenever, for any reason whatsoever, there is a polarization \(\textbf P\neq\textbf 0\) (often when an \(\textbf E_{\text{ext}}\) is applied such as in Problems #\(1\) and #\(2\), though see ferroelectrics), then there must necessarily be a build-up of electric charge somewhere in a material volume \(V\) (or on its surface \(\partial V\)); any such charge is called bound charge. This is because the electrostatic potential \(\phi\) induced by this \(\textbf P\) would be given by:

\[\phi(\textbf x)=\frac{1}{4\pi\varepsilon_0}\iiint_{\textbf x’\in V}\frac{\textbf P(\textbf x’)\cdot(\textbf x-\textbf x’)}{|\textbf x-\textbf x’|^3}d^3\textbf x’\]

But noting that (e.g. in spherical coordinates) \(\frac{\partial}{\partial\textbf x’}\frac{1}{|\textbf x-\textbf x’|}=\frac{\textbf x-\textbf x’}{|\textbf x-\textbf x’|^3}\), an integration by parts shows that:

\[\phi(\textbf x)=\frac{1}{4\pi\varepsilon_0}\iint_{\textbf x’\in\partial V}\frac{\sigma_b(\textbf x’)}{|\textbf x-\textbf x’|}d^2\textbf x’+\frac{1}{4\pi\varepsilon_0}\iiint_{\textbf x’\in V}\frac{\rho_b(\textbf x’)}{|\textbf x-\textbf x’|}d^3\textbf x’\]

(note that right now both of these are only electrostatic results because they arise by comparison to Coulomb’s law). When \(\textbf P\) is uniformly polarized, then bound charge does not clump up anywhere inside the material \(\rho_b=0\), but only on its surface \(\sigma_b\neq 0\).

Problem #\(4\): Explain what the concept of free charge is, and show that for a linear dielectric sphere of radius \(R\) with a lump of charge \(Q_f\) deposited at its center, the surface bound charge density is \(\sigma_b=\frac{\chi_e Q_f}{4\pi R^2(1+\chi_e)}\). What is the corresponding discontinuity \(\Delta\textbf E\) in the \(\textbf E\)-field across the surface of the linear dielectric sphere?

Solution #\(4\): Free charge is any charge that has clumped somewhere in a material, but that is not due to polarization \(\textbf P\), i.e. it is not bound charge. Essentially, it is a wastebasket for non-bound charge so that \(\rho=\rho_f+\rho_b\). Defining the displacement field as \(\textbf D:=\varepsilon_0\textbf E+\textbf P\) so that \(\frac{\partial}{\partial\textbf x}\cdot\textbf D=\rho_f\), it follows that for the free charge distribution \(\rho_f(\textbf x)=Q_f\delta^3(\textbf x)\), one has (everywhere in space \(\textbf x\in\textbf R^3-\{\textbf 0\}\), not just inside the dielectric sphere!):

\[\textbf D(\textbf x)=\frac{Q_f}{4\pi|\textbf x|^2}\hat{\textbf x}\]

Outside the dielectric sphere \(|\textbf x|\geq R\) where \(\textbf P=\textbf 0\), this reproduces the usual electric field \(\textbf E(\textbf x)=Q_f/4\pi\varepsilon_0|\textbf x|^2\hat{\textbf x}\). Meanwhile, for \(|\textbf x|\leq R\), the assumption of dielectric linearity allows one to write \(\textbf D=\varepsilon\textbf E\) with the permittivity \(\varepsilon=\varepsilon_0(1+\chi_e)\), yielding an internal electric field \(\textbf E(\textbf x)=Q_f/4\pi\varepsilon|\textbf x|^2\hat{\textbf x}\) screened by bound charge, and a corresponding induced polarization:

\[\textbf P(\textbf x)=\frac{\chi_e Q_f}{4\pi(1+\chi_e)|\textbf x|^2}\hat{\textbf x}\]

The claim then follows from \(\sigma_b=\hat{\textbf x}\cdot\textbf P(|\textbf x|=R)\). Alternatively, the bound charge \(\rho_b(\textbf x)=Q_b\delta^3(\textbf x)\) polarized towards the origin is clearly (using \(\rho_b=-\partial/\partial\textbf x\cdot\textbf P\) and the divergence theorem) \(Q_b=-\chi_eQ_f/(1+\chi_e)\) so that the net charge at the origin is \(Q_f+Q_b=Q_f/(1+\chi_e)\), consistent with the form of the \(\textbf E\)-field inside the linear dielectric sphere. But since the sphere as a whole should only contain the central free charge \(Q_f\), this requires it to have surface charge \(-Q_b\) from which \(\sigma_b=-Q_b/4\pi R^2\) leads to the same answer.

Although the \(\textbf D\)-field is continuous \(\Delta\textbf D=\textbf 0\) across the sphere surface (from \(\frac{\partial}{\partial\textbf x}\cdot\textbf D=\rho_f\) and a Gaussian pillbox, this is seen to be due to the lack of free surface charges \(\sigma_f=0\)), the \(\textbf E\)-field discontinuity is \(\Delta\textbf E=\sigma_b/\varepsilon_0\).

Problem #\(5\): Derive the analog of Problem #\(3\) for magnetostatics.

Solution #\(5\): Everything is completely analogous, in particular the analog of \(\phi\) is \(\textbf A\):

\[\textbf A(\textbf x)=\frac{\mu_0}{4\pi}\iiint_{\textbf x’\in V}\frac{\textbf M(\textbf x’)\times(\textbf x-\textbf x’)}{|\textbf x-\textbf x’|^3}d^3\textbf x’\]

Applying the same identity and tricks as in Solution #\(3\):

\[\textbf A(\textbf x)=\frac{\mu_0}{4\pi}\iint_{\textbf x’\in\partial V}\frac{\textbf K_b(\textbf x’)}{|\textbf x-\textbf x’|}d^2\textbf x’+\frac{\mu_0}{4\pi}\iiint_{\textbf x’\in V}\frac{\textbf J_b(\textbf x’)}{|\textbf x-\textbf x’|}d^3\textbf x’\]

where the bound surface current is \(\textbf K_b=\textbf M\times\hat{\textbf n}\) and the bound current is \(\textbf J_b=\frac{\partial}{\partial\textbf x}\times\textbf M\) (note that right now both of these are only magnetostatic results because they arise by comparison to the Biot-Savart law).

Problem #\(6\): Unfortunately there is no simple classical analog of Problem #\(1\) that can be used to rationalize why it is the case that many materials are linear not just as dielectrics, but also as diamagnets/paramagnets (quantum mechanics is needed!). So for now, taking as an experimental fact that \(\textbf M_{\text{ind}}=\chi_m\textbf H_{\text{ext}}\) for some magnetic susceptibility \(\chi_m\in\textbf R\), explain why \(\mu=\mu_0(1+\chi_m)\) (cf. \(\varepsilon=\varepsilon_0(1+\chi_e)\)).

Solution #\(6\): Most of the time when one thinks of “current”, one is thinking of free current \(\textbf J_f\) which contributes to the magnetic field \(\textbf B\) on top of any bound current \(\textbf J_b\) due to magnetization \(\textbf M\neq\textbf 0\). Ampere’s law of magnetostatics asserts that:

\[\frac{\partial}{\partial\textbf x}\times\textbf B=\mu_0(\textbf J_f+\textbf J_b)\]

So this motivates the definition of the magnetizing field \(\textbf H:=\textbf B/\mu_0-\textbf M\) so that \(\partial/\partial\textbf x\times\textbf H=\textbf J_f\). That being said, analogous to how \(\textbf D=\varepsilon\textbf E\) in linear dielectrics, one would also like \(\textbf H=\textbf B/\mu\) in these “linear diamagnets/paramagnets”. The claim follows.

Problem #\(7\): Explain what changes of the electrostatic/magnetostatic results above when the fields are taken to depend on time \(t\).

Solution #\(7\): Just as Maxwell’s displacement current contribution to Ampere’s original magnetostatic law came about by enforcing local charge conservation, here the idea will be to enforce local bound charge conservation, with local free charge conservation therefore coming as a corollary. As usual, this means stipulating the continuity equation:

\[\frac{\partial\rho_b}{\partial t}+\frac{\partial}{\partial\textbf x}\cdot\textbf J_b=0\]

The earlier magnetostatic result \(\textbf J_b=\partial/\partial\textbf x\times\textbf M\) is clearly not compatible with this if \(\dot{\rho}_b\neq 0\) is changing, and therefore, because \(\rho_b=-\partial/\partial\textbf x\cdot\textbf P\), needs to be corrected to:

\[\textbf J_b=\frac{\partial}{\partial\textbf x}\times\textbf M+\frac{\partial\textbf P}{\partial t}\]

Problem #\(8\): Write down Maxwell’s equations in materials. Comment on the case \(\rho_f=0\) and \(\textbf J_f=\textbf 0\).

Solution #\(8\):

\[\frac{\partial}{\partial\textbf x}\cdot\textbf D=\rho_f\]

\[\frac{\partial}{\partial\textbf x}\cdot\textbf B=0\]

\[\frac{\partial}{\partial\textbf x}\times\textbf E=-\frac{\partial\textbf B}{\partial t}\]

\[\frac{\partial}{\partial\textbf x}\times\textbf H=\textbf J_f+\frac{\partial\textbf D}{\partial t}\]

These should always be viewed with in conjunction with a pair of constitutive relations, usually just the linear approximations \(\textbf D=\varepsilon\textbf E\) and \(\textbf H=\textbf B/\mu\). In dielectrics where \(\rho_f=0\) and \(\textbf J_f=\textbf 0\), Maxwell’s equations look like the usual ones in vacuum but with \(\varepsilon_0\mapsto\varepsilon\) and \(\mu_0\mapsto\mu\) (assuming the dielectric is linear). Light waves can therefore propagate in linear dielectrics just as they would in vacuum, except at a slower speed \(v=(\mu\varepsilon)^{-1/2}<c\).

Problem #\(9\): Using the results of Solution #\(8\), derive the \(4\) electromagnetic interface conditions (hint: each of the \(4\) macroscopic Maxwell equations will provide one).

Solution #\(9\): “Use pillboxes for the divs and closed loops for the curls”:

\[\Delta\textbf D\cdot\hat{\textbf n}=\sigma_f\]

\[\Delta\textbf B\cdot\hat{\textbf n}=0\]

\[\hat{\textbf n}\times\Delta\textbf E=\textbf 0\]

\[\hat{\textbf n}\times\Delta\textbf H=\textbf K_f\]

Problem: Consider placing a thin rod aligned with an external magnetic field \(\textbf B_{\text{ext}}\). What is the internal magnetic field \(\textbf B_{\text{int}}\) and magnetizing field \(\textbf H_{\text{int}}\)?

Solution:

Problem: Repeat the above problem but for a thin slab.

Solution: Following the discussion above, the dominant boundary condition is the normal continuity of the \(\textbf B\)-field:

Problem #\(10\): Derive the law of reflection and Snell’s law.

Solution #\(10\): Applying phase-matching at the \(z=0\) interface (otherwise there could be no hope of satisfying the electromagnetic interface conditions), one concludes that the frequency is solely a property of the incident source \(\omega=\omega’=\omega^{\prime\prime}\), reflecting conservation of photon energy, and that the projections \(k\sin\theta=k’\sin\theta’=k^{\prime\prime}\sin\theta^{\prime\prime}\) of the wavevectors \(\textbf k,\textbf k’\) and \(\textbf k^{\prime\prime}\) onto the \(xy\)-plane \(z=0\) are also equal, reflecting conservation of photon momentum along that direction. The law of reflection then follows simply because the reflected ray is in the same dielectric \(n^{\prime\prime}=n\) as the incident ray, while Snell’s law \(n\sin\theta=n’\sin\theta’\) follows because the refracted ray is now in a different dielectric \(n’\neq n\) with different dispersion relation.

(aside: provided the media actually support wave propagation and that all indices of refraction are defined via the phase velocity so that \(n,n’\) could be less than \(1\), then both the law of reflection and Snell’s law hold at the interface of arbitrary media, not just two dielectrics. Furthermore, both laws also hold for water waves and sound waves, not just light waves, which emphasizes that they are not really corollaries of Maxwell’s equations, but corollaries of the wave equation, while for light waves the wave equation is a corollary of Maxwell).

Problem #\(11\): Using the result of Problem #\(10\), derive the Fresnel equations for reflection and transmission at dielectric interfaces.

Solution #\(11\): The assumption of dielectrics is needed here in order to set \(\sigma_f=0\) and \(\textbf K_f=\textbf 0\) in the electromagnetic interface conditions (i.e. would not be true for conductors!). Note also that the amplitudes \(\textbf E_0,\textbf E’_0,\textbf E^{\prime\prime}_0\) are arbitrary, so capture all cases of linear, circular, or elliptical polarization.

The incident wavevector is \(\textbf k=k\sin\theta\hat{\boldsymbol{\rho}}_{\phi}+k\cos\theta\hat{\textbf z}\) while the transmitted wavevector is \(\textbf k’=k’\sin\theta’\hat{\boldsymbol{\rho}}_{\phi}+k’\cos\theta’\hat{\textbf z}\); the incident electric field amplitude \(\textbf E_0\) lives in the orthogonal complement \(\text{span}^{\perp}(\textbf k)\) while \(\textbf E’_0\) lives in \(\text{span}^{\perp}(\textbf k’)\) (analogous remarks apply for the magnetic field amplitudes \(\textbf B_0,\textbf B’_0\) which are inseparably married to their respective electric fields but are not indicated on the diagram for simplicity). These two-dimensional subspaces admit convenient orthonormal bases given by the normal unit vector \(\hat{\textbf s}:=\hat{\textbf k}\times\hat{\textbf p}=\hat{\textbf k}’\times\hat{\textbf p}’\) and the parallel unit vectors \(\hat{\textbf p}:=\text{cot}\theta\hat{\textbf k}-\text{csc}\theta\hat{\textbf z}\) and likewise \(\hat{\textbf p}’:=\text{cot}\theta’\hat{\textbf k}’-\text{csc}\theta’\hat{\textbf z}\); here the words “normal” and “parallel” are with respect to the plane of incidence/transmission/reflection \(\text{span}(\textbf k,\hat{\textbf z})=\text{span}(\textbf k’,\hat{\textbf z})\).

By decomposing \(\textbf E_0=E_{0,s}\hat{\textbf s}+E_{0,p}\hat{\textbf p}\) where \(E_{0,s}=\textbf E_0\cdot\hat{\textbf s}\) and \(E_{0,p}=\textbf E_0\cdot\hat{\textbf p}\), and doing likewise for \(\textbf E’_0,\textbf E^{\prime\prime}_0,\textbf B_0,\textbf B’_0,\textbf B^{\prime\prime}_0\) and substituting all this into the \(4\) electromagnetic interface conditions with \(\hat{\textbf n}=\hat{\textbf z}\), one obtains the Fresnel equations:

\[t_s:=\frac{E’_{0,s}}{E_{0,s}}=\frac{2Y\cos\theta}{Y\cos\theta+Y’\cos\theta’}\]

\[r_s:=\frac{E^{\prime\prime}_{0,p}}{E_{0,p}}=\frac{Y\cos\theta-Y’\cos\theta’}{Y\cos\theta+Y’\cos\theta’}\]

\[t_p:=\frac{E’_{0,p}}{E_{0,p}}=\frac{2Y\cos\theta}{Y\cos\theta’+Y’\cos\theta}\]

\[r_p:=\frac{E^{\prime\prime}_{0,p}}{E_{0,p}}=\frac{Y\cos\theta’-Y’\cos\theta}{Y\cos\theta’+Y’\cos\theta}\]

where the admittances are \(Y:=Z^{-1}=\sqrt{\varepsilon/\mu}\) and likewise \(Y’:=Z’^{-1}=\sqrt{\varepsilon’/\mu’}\). Note here that all \(4\) coefficients \(t_s,r_s,t_p,r_p\) are defined here with respect to the electric field; one could have also defined them from the \(s\) and \(p\)-polarized components of the magnetic field, leading to almost the same Fresnel equations (differing by some subtle minus signs here and there).

Problem #\(12\): Sketch graphs of \(t_s,r_s,t_p,r_p\) as a function of the incident angle \(\theta\) in the two cases \(Y>Y’\) and \(Y<Y’\), and comment.

Solution #\(12\): From air to glass \(Y’/Y\approx 1.5\) (assuming both have \(\mu=\mu’=\mu_0\)):

At normal incidence \(\theta=\theta’=0\), there is clearly no longer any distinction between \(s\) and \(p\)-polarization, reflecting the fact that (as is clear from the graph) \(t_s(\theta=0)=t_p(\theta=0)\) and \(r_s(\theta=0)=r_p(\theta=0)\) regardless of whether \(Y<Y’\) or \(Y>Y’\). In addition, again regardless of whether \(Y<Y’\) or \(Y>Y’\), there always exists a Brewster angle \(\tan\theta_B=\sqrt{\frac{1-(Y’/Y)^2}{(n/n’)^2-1}}\approx Y’/Y\) (approximation valid for non-magnetic dielectrics e.g. air, glass for which \(Y’/Y\approx n’/n\) and the reflected light \(\textbf k’\cdot\textbf k^{\prime\prime}=0\) is orthogonal to the refracted light \(\theta_B+\theta’_B\approx \pi/2\)) and is entirely linearly \(s\)-polarized since \(r_p(\theta=\theta_B)=0\). However, only in the case \(Y>Y’\) of going from an optically dense to less dense medium does there exist a critical angle \(\theta_C=\arcsin(Y’/Y)\) beyond which \(\theta>\theta_C\) only total internal reflection occurs together with transmission of an evanescent wave \(\textbf E'(\rho,\phi,z)=\textbf E’_0e^{-\sqrt{k^2\sin^2\theta-k’^2}z}e^{i(k\rho\cos(\phi-\phi_{\textbf k})\sin\theta-\omega t)}\).

Problem #\(13\): Building off of Problem #\(12\), calculate the transmitted powers \(T_s(\theta),T_p(\theta)\) along with the reflected powers \(R_s(\theta),R_p(\theta)\) as a function of incident angle \(\theta\) for both \(s\) and \(p\)-polarization. Verify that \(T_s(\theta)+R_s(\theta)=T_p(\theta)+R_p(\theta)=1\). Show that for air-to-glass at normal incidence (so that there is no distinction between \(s\) and \(p\)-polarization), \(R(\theta=0)\approx 4\%\).

Solution #\(13\): As long as one remembers the expression for the (period-averaged) Poynting vector \(\textbf S=\frac{1}{2}\textbf E\times\textbf H^*=\frac{Y|\textbf E_0|^2}{2}\hat{\textbf k}\), then the rest is easy (by definition the coefficients here are defined relative to the normal \(\hat{\textbf n}=\hat{\textbf z}\) of the interface though Poynting’s theorem with \(\textbf J_f=\textbf 0\) would hold in an arbitrary direction; there is also a nice way to see why this is using notions of “beam divergence”).

\[T_s(\theta)=\frac{\textbf S’_s\cdot\hat{\textbf z}}{\textbf S_s\cdot\hat{\textbf z}}=\frac{Y’\cos\theta’}{Y\cos\theta}|t_s(\theta)|^2\]

\[R_s(\theta)=\frac{\textbf S^{\prime\prime}_s\cdot\hat{\textbf z}}{\textbf S_s\cdot\hat{\textbf z}}=|r_s(\theta)|^2\]

\[T_p(\theta)=\frac{\textbf S’_p\cdot\hat{\textbf z}}{\textbf S_p\cdot\hat{\textbf z}}=\frac{Y’\cos\theta’}{Y\cos\theta}|t_p(\theta)|^2\]

\[R_p(\theta)=\frac{\textbf S^{\prime\prime}_p\cdot\hat{\textbf z}}{\textbf S_p\cdot\hat{\textbf z}}=|r_p(\theta)|^2\]

the rest is easy to check (aside: given the \(4\) electromagnetic boundary conditions earlier, can one write down some corresponding boundary conditions on \(\hat{\textbf n}\cdot\Delta\textbf S\) and \(\hat{\textbf n}\times\Delta\textbf S\) which capture energy conservation? Especially the first one seems to just coming from Poynting’s theorem).

Problem #\(14\): Using the Lorentz oscillator model of the electron \(e^-\), show how the atomic polarizability \(\alpha\), the electric susceptibility \(\chi_e\), the permittivity \(\varepsilon\), and the wavenumber \(k\) all become complex-valued functions of the frequency \(\omega\in\textbf R\) of the incident light.

Solution #\(14\): The equation of motion for the electron is taken to be of the usual form of a damped, driven harmonic oscillator:

\[\ddot{\textbf x}(t)+\Delta\omega\dot{\textbf x}(t)+\omega_0^2\textbf x(t)=\frac{q\textbf E_0}{m}e^{i(\textbf k_{\omega}\cdot\textbf x(t)-\omega t)}\]

where the damping coefficient \(\Delta\omega\) and the resonant frequency \(\omega_0\) are both phenomenological parameters of the Lorentz model (the former \(\Delta\omega\) is the quantum line width of the relevant optical transitions that have energy commensurate with \(\hbar\omega\), equivalently the spontaneous emission rate or reciprocal lifetime, while the latter \(\omega_0^2=q^2/4\pi\varepsilon_0ma^3\) has a classical form following Solution #\(1\)). Although of course the electric field \(\textbf E(\textbf x,t)=\textbf E_0e^{i(\textbf k_{\omega}\cdot\textbf x-\omega t)}\) of course has an accompanying magnetic field \(\textbf B(\textbf x,t)=\textbf B_0e^{i(\textbf k_{\omega}\cdot\textbf x-\omega t)}\), the magnetic force \(q\dot{\textbf x}\times\textbf B\) can be ignored in the non-relativistic limit \(|\dot{\textbf x}|\ll c\).

Making the further assumption that at the relevant frequency \(\omega\), \(\textbf k_{\omega}\cdot\textbf x(t)\ll 1\) at all times \(t\), then it really reduces to a linearly damped, driven harmonic oscillator, for which the steady state particular integral oscillation is:

\[\textbf x=\frac{q}{m}\frac{1}{\omega_0^2-\omega^2-i\omega\Delta\omega}\textbf E\]

Optionally multiplying this solution \(\textbf x\) by the electron charge \(q\), one can write the electric dipole moment of an atom as:

\[\textbf p=\frac{q^2}{m}\frac{1}{\omega_0^2-\omega^2-i\omega\Delta\omega}\textbf E\]

But from the definition \(\textbf p=\alpha\textbf E\), one immediately recognizes the atomic polarizability has become a \(\textbf C\)-valued function of \(\omega\in\textbf R\):

\[\alpha(\omega)=\frac{q^2}{m}\frac{1}{\omega_0^2-\omega^2-i\omega\Delta\omega}\]

From \(\textbf p\), one can get \(\textbf P\) through a factor of the number density \(n\) of electrons (equivalent to the number density of electric dipole moments):

\[\textbf P=n\textbf p=n\alpha\textbf E:=\varepsilon_0\chi_e\textbf E\]

So:

\[\chi_e(\omega)=\frac{\omega_p^2}{\omega_0^2-\omega^2-i\omega\Delta\omega}\]

where the plasma frequency \(\omega_p^2=nq^2/m\varepsilon_0\) scales with the plasma density \(n\). The \(\textbf C\)-valued nature of the permittivity is then evident from \(\varepsilon(\omega)=\varepsilon_0(1+\chi_e(\omega))\) whose the real and imaginary parts are:

\[\Re\varepsilon(\omega)=\varepsilon_0\left(1-\frac{\omega_p^2(\omega^2-\omega_0^2)}{(\omega^2-\omega_0^2)^2+\omega^2\Delta\omega^2}\right)\]

\[\Im\varepsilon(\omega)=\varepsilon_0\frac{\omega_p^2\omega\Delta\omega}{(\omega^2-\omega_0^2)^2+\omega^2\Delta\omega^2}\]

And the shape of both of their graphs is a good thing to get familiar with (note the roles played by the \(3\) distinct frequency scales in the function \(\omega_p,\omega_0,\Delta\omega\)):

As an aside, because the complexified \(\varepsilon(\omega)\) is complex analytic in the upper half-plane \(\Im\omega>0\) on causality grounds, its real and imaginary parts \(\Re\varepsilon(\omega),\Im\varepsilon(\omega)\) are related by Kramers-Kronig relations:

\[\Re\varepsilon(\omega)=\mathcal P\int_{-\infty}^{\infty}\frac{d\omega’}{\pi}\frac{\Im\varepsilon(\omega’)}{\omega’-\omega}\]

\[\Im\varepsilon(\omega)=\mathcal P\int_{-\infty}^{\infty}\frac{d\omega’}{\pi}\frac{\Re\varepsilon(\omega’)}{\omega-\omega’}\]

Problem #\(15\): Dispersion relations for waves are conventionally thought of as \(\omega=\omega_{\textbf k}\), but one can of course invert this to obtain \(\textbf k=\textbf k_{\omega}\) and view that as the dispersion relation. This view will be more useful here; think of the frequency \(\omega\in\textbf R\) as free real parameter one gets to select as the experimentalist.

For a linear dielectric in the absence of (free) charges or (free conduction) currents, what consequences does \(\varepsilon(\omega)\in\textbf C\) have for the propagation of light in such a medium?

Solution #\(15\): One always has the “naive linear” nondispersion:

\[|\textbf k_{\omega}|=\sqrt{\mu\varepsilon}\omega\]

However, even though one typically assumes \(\mu\in\textbf R\) to be \(\omega\)-independent, the fact that \(\varepsilon=\varepsilon(\omega)\in\textbf C\) promotes the wavevector \(\textbf k_{\omega}\in\textbf C^3\) rather than the usual \(\textbf k_{\omega}\in\textbf R^3\). There are \(2\) important consequences of this, illustrated for instance using \(\textbf B(\textbf x,t)\):

Problem #\(16\): Show using the Drude model that all of the discussion for dielectrics applies equally well to conductors provided the permittivity receives an additional \(\textbf C\)-contribution from conduction electrons of the form:

\[\varepsilon(\omega)\mapsto\varepsilon_{\text{eff}}(\omega):=\varepsilon(\omega)+\frac{i\sigma(\omega)}{\omega}\]

where the optical conductivity is given by the Mobius transformation:

\[\sigma(\omega)=\frac{\sigma_{\text{DC}}}{1-i\omega\tau}\]

where \(\sigma_{\text{DC}}=\sigma(\omega=0)=nq^2\tau/m=\varepsilon_0\tau\omega_p^2\) and \(\tau>0\) is the relaxation time between electron collisions.

Solution #\(16\): Now the equation of motion is basically the same as from the Lorentz oscillator model except there is no Hookean restoring force \(\omega_0=0\) and one renames \(\Delta\omega=1/\tau\). Notably, because the ODE is now merely \(1^{\text{st}}\)-order:

\[\dot{\textbf v}+\frac{1}{\tau}\textbf v=\frac{q\textbf E}{m}\]

The steady state particular integral velocity response \(\textbf v(t)=\textbf v_0e^{-i\omega t}\) to an AC electric field \(\textbf E=\textbf E_0e^{-i\omega t}\) is now a Mobius rather than a Lorentzian:

\[\textbf v=\frac{q\tau}{m}\frac{1}{1-i\omega\tau}\textbf E\]

In order to get optical conductivity \(\sigma(\omega)\), one just has to remember what its defining equation is in the first place, i.e. Ohm’s law \(\textbf J=\sigma\textbf E\). So computing \(\textbf J=\rho\textbf v=\frac{nq^2\tau}{m}\frac{1}{1-i\omega\tau}\textbf E\), the claim follows.

In Fourier space, applying Ohm’s law (valid for low mean free path of the electrons) and the continuity equation for local charge conservation, Maxwell’s equations in an ohmic conductor:

\[i\textbf k_{\omega}\cdot\varepsilon\textbf E_0=\frac{\textbf k_{\omega}\cdot\sigma\textbf E_0}{\omega}\]

\[i\textbf k_{\omega}\cdot\textbf B_0=0\]

\[i\textbf k_{\omega}\times\textbf E_0=-(-i\omega\textbf B_0)\]

\[i\textbf k_{\omega}\times\frac{\textbf B}{\mu}=\sigma\textbf E_0-i\omega\varepsilon\textbf E_0\]

And in particular, the last equation (Ampere-Maxwell) gives the desired \(\varepsilon_{\text{eff}}=\varepsilon+i\sigma/\omega\) (interestingly this same quantity is required to be non-vanishing \(\varepsilon_{\text{eff}}\neq 0\) so that Gauss’s law for the \(\textbf D\)-field yields transverse EM waves). Computing the usual \(\textbf k_{\omega}\times(\textbf k_{\omega}\times\textbf E_0)\) gives the dispersion relation:

\[k_{\omega}=\sqrt{\mu\varepsilon_{\text{eff}}}\omega\]

Problem #\(17\): How do waves behave in the low-\(\omega\) and high-\(\omega\) limits inside the conductor?

Solution #\(17\): For low-\(\omega\), the conductivity term dominates so the permittivity \(\varepsilon_{\text{eff}}\approx\varepsilon_0\frac{i\sigma}{\omega}\approx\varepsilon_0\frac{i\sigma_{\text{DC}}}{\omega}\) is purely imaginary. The dispersion relation then dictates:

\[k_{\omega}=\sqrt{\mu\varepsilon_{\text{eff}}}\omega=\sqrt{\frac{\mu\sigma_{\text{DC}}\omega}{2}}(1+i)\]

where the other option \(\sqrt{i}=-(1+i)/\sqrt{2}\) is rejected because one requires on physical grounds an evanescent wave so \(\Im k_{\omega}>0\). Moreover, because \(\Re k_{\omega}=\Im k_{\omega}\) in this case, the \(\textbf E\) (equivalently, roughly \(\textbf D\)) and \(\textbf B\) (equivalently \(\textbf H\)) fields are \(45^{\circ}\) out of phase in the conductor. All of the EM fields \(\textbf E,\textbf B,\textbf D,\textbf H\) are exponentially attenuated:

\[(\text{phasor})e^{i(\textbf k_{\omega}\cdot\textbf x-\omega t)}=(\text{phasor})e^{-z/\delta(\omega)}e^{i(z/\delta(\omega)-\omega t)}\]

where the skin depth \(\delta(\omega):=\sqrt{2/\mu\sigma_{\text{DC}}\omega}=\lambda/2\pi\) is on the order of a single wavelength \(\lambda\)!

For high-\(\omega\), \(\varepsilon\to\varepsilon_0(1-\omega_p^2/\omega^2)\) and \(\sigma\to i\sigma_{\text{DC}}/\omega\tau\) (just look at the graphs from earlier to see all of this!) so:

\[\varepsilon_{\text{eff}}\to\varepsilon_0\left(1-\frac{2\omega_p^2}{\omega_0^2}\right)\in\textbf R\]

The implication of this is that \(\omega_p\) (or more precisely \(\sqrt{2}\omega_p\)) provides a strict cut-off:

The atmosphere can be modelled as such a plasma and this underlies why FM radio waves with \(\omega>\sqrt{2}\omega_p\) transmit through the atmosphere whereas AM radio waves with \(\omega<\sqrt{2}\omega_p\) are reflected.

Problem: Explain how to make a Zener diode and the physics underlying its breakdown mechanism.

Solution: A Zener diode is simply made by heavily doping a \(p\)-\(n\) junction, leading to a very thin depletion region. Then, upon applying a suitably strong reverse bias, eventually electrons from the valence band of the \(\text{p}\)-type semiconductor can quantum tunnel into the conduction band of the \(\text n\)-type semiconductor, hence giving rise to Zener breakdown; the Zener voltage at which this occurs is typically \(V_{\text{Zener}}<5\text{ V}\) or so. Beyond this point, a substantial generation current flows across the \(\text p\)-\(\text n\) junction.

Problem: Explain how to make an avalanche diode and the physics underlying its breakdown mechanism.

Solution: The avalanche diode is sometimes also marketed as a Zener diode simply because in practical electronics, both serve the same function, e.g. voltage referencing, protection in flyback Zener diodes, etc.

This time, the idea is to lightly dope instead. One thus gets a large depletion region. Minority mobile charge carriers accelerated across the depletion region then have enough energy to ionize core electrons, leaving behind holes—>3 carriers, each of which is further accelerated by the \(\textbf E\)-field, and can ionize more Si host atoms. This is thus an avalanche effect, leading to avalanche breakdown (e.g. in single-photon avalanche diodes).

Problem: A smartphone tuning app is able to tune the fifth string of a guitar to \(110\text{ Hz}\) with a precision of \(0.07\text{ Hz}\). Estimate the minimum sampling frequency and
sampling time needed for this task.

Solution:

\[f_s\geq 2\times(110+0.07)\text{ Hz}=220.14\text{ Hz}\]

\[T_s\geq\frac{1}{0.07\text{ Hz}}\approx 14\text{ s}\]

Digitization & Signal Processing

In what follows, it is convenient to think of \(f(t)\in\textbf R\) as a real-valued analog \(t\)-domain signal. Then its Fourier transform \(\hat f(\omega):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-i\omega t}dt\) (exact convention doesn’t matter) is trivially seen to be conjugate-even in \(\omega\)-space \(\hat f(-\omega)=\hat f^*(\omega)\) and in particular its modulus \(|\hat f(\omega)|\) (which one often plots and speaks of as if it were \(\hat f(\omega)\) itself) is simply an even function in \(\omega\)-space \(|\hat f(-\omega)|=|\hat f(\omega)|\). Thus, as Newton might say, for every positive frequency component \(\omega>0\) in \(\hat f(\omega)\), there is an “equal and opposite” negative frequency component \(-\omega<0\) in \(\hat f(\omega)\). This notion that \(\hat f(\omega)\) will be symmetric around the origin \(\omega=0\) is important to keep in mind in what follows below:

(Weak Nyquist Theorem): Given a \(t\)-domain signal \(f(t)\) whose Fourier transform \(\hat f(\omega)\) is known a priori to be compactly supported (or bandlimited) in \(\omega\)-space on a symmetric interval of the form \([-\omega^*,\omega^*]\), then one can guarantee recovery of \(f(t)\) from the \(t\)-sampled signal \(f(t)Ш_{2\pi/\omega_s}(t)\) at sampling angular frequency \(\omega_s\) (where the Dirac comb is defined by \(Ш_{T}(t):=\sum_{n=-\infty}^{\infty}\delta(t-nT)\)) provided one \(t\)-samples with at least \(\omega_s>2\omega^*\).

Proof: Given the \(t\)-sampled signal \(f(t)Ш_{2\pi/\omega_s}(t)\) at sampling angular frequency \(\omega_s\), the Fourier transform of this product is (by the convolution theorem) the convolution of their Fourier transforms (where the Dirac comb is roughly speaking an eigenfunction of the Fourier transform):

\[f(t)Ш_{2\pi/\omega_s}(t)\mapsto\hat f(\omega)*Ш_{\omega_s}(\omega)\]

which is also equal to \(\hat f(\omega)*Ш_{\omega_s}(\omega)=\sum_{n=0}^{\infty}\hat f(\omega-n\omega_s)\), i.e. placing translated copies of the graph of \(\hat f(\omega)\) at each discrete lattice point \(n\omega_s\) in \(\omega\)-space. No information is lost from sampling if and only if these so-called convolution images \(\hat f(\omega-n\omega_s)\) for \(n\in\textbf Z\) do not overlap with each other in \(\omega\)-space for which it is a sufficient (though not necessary) condition that \(\omega_s> 2\omega^*\) as claimed. More precisely, this claimed losslessness of information is captured by the following explicit algorithm for recovering this information: first, multiplying \(\hat f(\omega)*Ш_{\omega_s}(\omega)\) by the top-hat filter \(\text{rect}(\omega/2\omega^*):=[\omega\in[-\omega^*,\omega^*]]\) in \(\omega\)-space filters out all the duplicate convolution images \(\hat f(\omega-n\omega_s)\) for \(n\neq 0\) arising from the \(t\)-sampling and leaves one with the desired, bandlimited Fourier transform of just the original, unsampled, signal \(f\):

\[\hat f(\omega)=[\hat f(\omega)*Ш_{\omega_s}(\omega)]\text{rect}(\omega/2\omega^*)\]

To recover the original \(t\)-domain signal \(f(t)\), the inverse Fourier transform leads (by the convolution theorem again) to:

\[f(t)=[f(t)Ш_{2\pi/\omega_s}(t)]*2\omega^*\text{sinc}\left(\omega^*t\right)\]

where \(\text{sinc}(x):=\sin(x)/x\) in this context is unnormalized \(\int_{-\infty}^{\infty}\text{sinc}(x)dx=\pi\) (see the Dirichlet integral; one can of course also work with the normalized \(\widehat{\text{sinc}}(x):=\text{sinc}(\pi x)\)) and acts as an interpolating kernel when convolved with the \(t\)-sampled signal \(f(t)Ш_{2\pi/\omega_s}(t)\) (this discussion therefore also proves the Whittaker-Shannon interpolation formula). Hence the information in \(f\) is conserved as promised.

However, given more information about \(f\) (specifically, about \(\hat f\)), one can accordingly do better. That is, suppose \(\hat f(\omega)\) is not merely supported on a symmetric interval in \(\omega\)-space of the form \([-\omega^*,\omega^*]\), but rather on a symmetric union of two intervals of the form \([-\omega^*,-\omega_*]\cup[\omega_*,\omega^*]\) with some lower angular frequency cutoff \(\omega_*<\omega^*\) known a priori in addition to the bandlimiting frequency \(\omega^*\). Then it turns out to be possible to recover \(f(t)\) from a sub-Nyquist \(t\)-sampling frequency \(\omega_s\leq 2\omega^*\) provided one chooses \(\omega_s\) wisely:

(Strong Nyquist Theorem): Given a \(t\)-domain signal \(f(t)\) whose Fourier transform \(\hat f(\omega)\) is known a priori to be bandlimited in \(\omega\)-space on a symmetric union of two intervals of the form \([-\omega^*,-\omega_*]\cup[\omega_*,\omega^*]\), then it may be possible to \(t\)-sample at a sub-Nyquist frequency \(\omega_s>2\Delta\omega\) (where \(\Delta\omega:=\omega^*-\omega_*\) is the bandwidth of the signal \(f\)) while still being able to recover \(f(t)\) from \(f(t)Ш_{2\pi/\omega_s}(t)\).

Proof: Just draw some pictures in \(\omega\)-space. Note the catch to this: such a sampling frequency \(\omega_s\) may not exist if there is not enough room to sneak aliases into the empty region \([-\omega_*,\omega_*]\). The exact nature of when this is possible are complicated:

In practice, for signals \(f(t)\) one encounters in the real world (e.g. temperature \(T(t)\) of a thermistor, air pressure \(p(t)\) of a microphone, displacement \(x(t)\) of a particle, etc. though in practice these are all reduced via modern electronics to voltages \(V(t)\) in a circuit), it is rare to find that \(\hat f(\omega)\) is localized precisely to some interval in \(\omega\)-space like \([-\omega^*,\omega^*]\) or \([-\omega^*,-\omega_*]\cup[\omega_*,\omega^*]\); usually the support \(\hat f^{-1}(\textbf C-\{0\})\) will be unbounded so that in this case the separate convolution images \(\hat f(\omega-n\omega_s)\) separated by the sampling frequency \(\omega_s\) could only be made disjoint from each other in the limit of infinite separation \(\omega_s\to\infty\) in \(\omega\)-space but of course there is no practical \(t\)-sampler with \(\omega_s=\infty\). However all is not lost; in many applications it turns out that one can think of \(\omega_*\) and \(\omega^*\) not as strictly the smallest and largest frequency components present in the signal \(\hat f(\omega)\), but rather as the smallest and largest frequency components in \(\hat f(\omega)\) that one gives a damn about for the application of interest. For instance, human biology limits our ears to being able to detect sound frequencies in between \(\omega_*/2\pi\approx 20\text{ Hz}\) and \(\omega^*/2\pi \approx 20\text{ kHz}\). Mathematically, this means that the signal \(f(t)\) corresponding to \(\hat f(\omega)\) and the signal \(\tilde f(t)\) corresponding to \(\hat f(\omega)\text{rect}_{2\Delta\omega}(\omega)\) sound exactly the same to our ears. The Nyquist theorem therefore asserts that one needs to sample at at least \(\omega_s/2\pi\geq\approx 40\text{ kHz}\) and indeed the standard audio sampling frequency turns out to be \(\omega_s/2\pi=44.1\text{ kHz}\).

Operational Amplifiers

An operational amplifier (op-amp for short) is, as its name suggests, a kind of amplifier. The “operational” part of its name refers to the fact that op-amp circuits can essentially be used to perform various mathematical operations on functions such as addition/subtraction \(\pm\), differentiation \(\frac{d}{dt}\), integration \(\int dt\), etc. The typical schematic of an op-amp is shown below (note there is no widespread convention regarding whether one should draw the non-inverting input \(+\) above or below the inverting input \(-\)):

Since op-amps are active components (unlike say resistors, capacitors or inductors which are passive components), they require a bipolar external DC power supply \(\pm V_S\) which is not always annotated on the schematic like it is above, being implicitly understood. Typically, but not always, \(V_S=15\text{ V}\). As far as practical use of op-amps goes, it suffices to treat them as black boxes (in reality lots of transistors, etc.) subject to the following \(3\) rules (strictly true only in the steady state \(t\to\infty\)):

\[V_{\text{out}}\leq V_s\]

\[V_{\text{out}}=A(V_+-V_-)\]

\[I_+=I_-=0\]

where \(A\) is called the op-amp’s open-loop gain because as it stands, the op-amp drawn above is said to be in an open-loop configuration meaning that there’s nothing at all connecting the output \(V_{\text{out}}\) to either of the inputs \(V_{\pm}\) (if there were, then the op-amp would be in a closed-loop configuration). The open-loop gain \(A\) is typically quite large \(A\sim 10^5\), and also not well-controlled during op-amp manufacturing (i.e. there can be a substantially large tolerance \(\Delta A\)). For this reason, despite the fact that the rule \(V_{\text{out}}=A(V_+-V_-)\) clearly suggests an amplification of the differential input voltage \(V_+-V_-\) by the open-loop gain \(A\), actually most of the time (with the exception of certain special op-amp circuits such as comparators), the amplifying ability of the op-amp is not meant to come from its open-loop gain \(A\). Put another way, if one tried to amplify even a mere \(V_+-V_-=0.2\text{ mV}\) using just the raw op-amp in its open-loop configuration, then \(A(V_+-V_-)\sim 20\text{ V}\) would already exceed the typical supply voltage \(V_S=15\text{ V}\), contradicting the first rule and therefore causing the op-amp in this open-loop configuration to simply saturate at \(V_{\text{out}}=V_S=15\text{ V}\).

So as it stands, we have some open-loop op-amp with some large open-loop gain \(A\) plagued by a large uncertainty \(\Delta A\) and only able to amplify the smallest of voltages before saturating. Doesn’t sound too useful! The key insight to addressing all of these problems is, as one may guess, to not be using open-loop op-amps, but rather to use op-amps in a closed-loop configuration. More precisely, there are \(2\) distinct kinds of closed-loop configuration, one generally more useful than the other. If one connects a wire from \(V_{\text{out}}\) to \(V_+\) (resp. \(V_-\)) (possibly with other components like resistors, capacitors, etc. along this wire), then the op-amp is said to be in a positive (resp. negative) feedback configuration. As its name suggests, a positive feedback configuration is self-reinforcing because, suppose initially \(V_+=V_-\) so that \(V_{\text{out}}=0\). Then, if some small perturbation causes a sudden imbalance \(V_+>V_-\), then of course \(V_{\text{out}}=A(V_+-V_-)\) will respond by increasing too. But now comes the positive feedback loop! Since \(V_{\text{out}}\sim V_+\) are correlated together in a positive feedback configuration, as \(V_{\text{out}}\) grows so does \(V_+\) which by \(V_{\text{out}}=A(V_+-V_-)\) causes \(V_{\text{out}}\) to grow even more, and so forth, leading to a runaway instability (in practice the op-amp would quickly rail at \(V_{\text{out}}=V_S\) and square-wave like oscillations of \(\pm V_S\) in \(V_{\text{out}}(t)\) may ensue). This is not very useful unless one is specifically interested in such an oscillator. By far the most useful closed-loop configuration of an op-amp is the negative feedback configuration. Repeating the above arguments shows that this tends to have a stabilizing effect by coaxing \(V_+-V-\to 0\) to a setpoint of \(0\) in a manner not unlike a mass on a spring which attempts to restore the mass’s position \(x\) to the equilibrium position \(x_0\) via \(x-x_0\to 0\). Heuristically:

Thus, for op-amps with a closed-loop negative-feedback configuration specifically, one can just assume that \(V_+\approx V_-\) even when \(V_{\text{out}}\neq 0\). This is called the virtual short rule. Together with the two other rules \(V_{\text{out}}\leq V_S\) and \(I_+=I_-=0\), these \(3\) rules taken together are called the golden rules for op-amps (sometimes the rule \(V_{\text{out}}\leq V_S\) is omitted from the golden rules but I think it’s important enough to be included among them). Note that the two golden rules \(V_{\text{out}}\leq V_S\) and \(I_+=I_-=0\) are always true, the former because of just how op-amps work (black box!) and the latter because the input impedance \(Z_+=Z_-=\infty\) is in practice very large, so effectively infinite on both inputs, hence no current is drawn \(I_-=I_+=0\) (this is clearly independent of whether the op-amp happens to be closed-loop or not). But the virtual short golden rule \(V_+=V_-\) only applies to op-amps in closed-loop negative-feedback configurations. This is a very important caveat that one must always remember. In other words, the first thing to always check before analyzing any op-amp circuit is whether or not there is a negative feedback path \(V_{\text{out}}\to V_-\). If so, then one’s life is made easy by the \(3\) golden rules. Otherwise, if the op-amp is in an open-loop or closed-loop but positive-feedback configuration, then one must be more careful and in particular one cannot just blindly apply the golden rule \(V_+=V_-\) anymore.

Perhaps the simplest closed-loop negative-feedback op-amp circuit is that of the voltage follower; one simply starts with an open-loop op-amp and then connects a negative feedback loop \(V_{\text{out}}\to V_-\) with absolutely nothing on it:

Thanks to the earlier discussion, one can therefore legitimately apply the golden rules here to find that, at equilibrium, \(V_{\text{out}}=V_+=V_-\), hence the name “voltage follower”. It may seem like a voltage follower is a bit of a pointless op-amp circuit because it simply “buffers” the input voltages \(V_+,V_-\) across to the output \(V_{\text{out}}\); after all a wire would do exactly the same thing. The key difference though is that whereas a wire will also draw some non-zero current \(I\neq 0\), a voltage follower, being essentially an op-amp, does not draw any current \(I_+=0\) as one of the op-amp golden rules. In other words, electrons \(e^-\) are allowed to hop between the two stages, so they are effectively isolated from each other and cannot affect each other. This point is best illustrated with the following explicit example in which the first stage is a voltage divider \((R,R)\) (same resistance \(R\) for simplicity) and the second stage is some load resistor \(R\) (again same \(R\) for simplicity). One would like to take the voltage outputted at the center of the voltage divider and apply it across the load resistor \(R\). Naively, applying an input voltage of \(V_{\text{in}}\) across the voltage divider leads to an output voltage \(V_{\text{out}}=V_{\text{in}}/2\). Suppose this is how much voltage we actually would like to apply across the load resistor \(R\). So then we connect the load resistor \(R\) to the voltage divider with a simple, naive wire:

the problem here of course is that the first stage (the voltage divider) is not isolated from the second stage (the load resistor \(R\)) because a current will flow across and into the load resistor \(R\). Because the load resistor \(R\) is in parallel with the bottom resistor \(R\) in the potential divider, one can check this implies that, rather than the voltage divider outputting the voltage \(V_{\text{in}}/2\) we had hoped for, actually it’s been reduced to \(V_{\text{in}}/3\) after we hooked up the load resistor \(R\). Of course in this case it’s not too much hassle to deal with this (just adjust \(V_{\text{in}}\) accordingly), but more generally it can be headache if each additional stage one adds messes with previous stages. Instead of connecting just a bare wire between the two stages, here a voltage follower comes in very handy:

Now, because \(I_+=0\), we recover our expected voltage divider output of \(V_{\text{in}}/2\). Moreover, the voltage follower then simply passes this voltage \(V_{\text{in}}/2\) across the second stage load resistor \(R\) without any fuss, as we desired. Voltage followers therefore provide a simple and effective way to buffer an output voltage from one stage as the input voltage into a second stage, ensuring that any modifications made to the second stage later (e.g. changing the resistance of the load resistor \(R\mapsto R’\neq R\)) do not affect this buffered voltage \(V_{\text{in}}/2\).

Earlier, it was mentioned that op-amps, while possessing amplification powers, are seldom used as open-loop amplifiers with their unstable open-loop gain \(A\). So then how do they amplify voltages? One way is to use an non-inverting amplifier. The idea is to start with the open-loop op-amp, then provide negative feedback through a feedback resistor \(R_F\), followed by completing the voltage divider with a second resistor \(R\) (of course, there are many topologically homeomorphic ways to draw this, in particular drawings that put the non-inverting input \(+\) above the inverting input \(-\)):

The fact that this is a voltage divider for \(V_{\text{out}}\) together with the golden rule (because we’re supplying negative feedback!) immediately yields \(V_{\text{in}}=\frac{R}{R+R_F}V_{\text{out}}\) from which we obtain the equation for a non-inverting amplifier:

\[V_{\text{out}}=\left(1+\frac{R_F}{R}\right)V_{\text{in}}\]

Notice that if one removes the feedback resistor \(R_F=0\), then the non-inverting amplifier reduces to a voltage follower. Notice also that \(1+\frac{R_F}{R}\geq 1\) is always amplifying, never attenuating.

Alternatively, there also exists the inverting amplifier, which derives from the non-inverting amplifier by simply swapping the locations of \(V_{\text{in}}\) and ground \(\text{GND}\):

In this case, equating the currents across the two resistors along the top wire gives the equation for an inverting amplifier:

\[V_{\text{out}}=-\frac{R_F}{R}V_{\text{in}}\]

the fact that \(\text{sgn}(V_{\text{out}})=-\text{sgn}(V_{\text{in}})\) explains why this is called an inverting amplifier (i.e. if \(V_{\text{in}}(t)\) were a time-dependent rather than just DC input voltage, then \(V_{\text{out}}(t)\) would also be time-dependent and \(\pi\) out of phase with \(V_{\text{in}}(t)\)). Actually, the non-inverting and inverting amplifiers together explain why the \(-\) input is called the inverting input in the first place, and likewise why the \(+\) input is called the non-inverting input (see where \(V_{\text{in}}\) is being inputted in the case of each amplifier circuit!). Unlike the non-inverting amplifier, the inverting amplifier can clearly either amplify or attenuate depending on the application. More generally, for any op-amp circuit there will always be some suitable notion of input voltage \(V_{\text{in}}\) along with the usual output voltage \(V_{\text{out}}\) so that for any op-amp circuit one can define its gain \(G:=V_{\text{out}}/V_{\text{in}}\). Thus, a voltage follower has \(G=1\) unit gain, a non-inverting amplifier has gain \(G=1+R_F/R\) while an inverting amplifier has negative gain \(G=-R_F/R\).

In any case, the important point here I want to emphasize again about these amplifier circuits is that, unlike the op-amp’s unstable intrinsic open-loop gain \(A\), these closed-loop gains \(G\) are essentially not functions of \(A\), i.e. \(\partial G/\partial A=0\) precisely because \(A\sim 10^5\) is so large. The point is that \(G\) is a much more controllable gain to work with for amplification purposes than \(A\) because it only depends on the values of external components like resistances \(R_F,R\) which are generally well-known with small tolerances. To hit this point home, consider the following abstraction of the process of negative feedback:

The op-amp is in a closed-loop negative-feedback configuration with a feedback factor \(\beta\in\textbf R\) (implemented through some external components, for instance \(\beta=-R/R_F\) in the inverting amplifier) defined so that it basically does what we intuitively expect negative feedback to do, namely \(V_-\mapsto V_{-}-\beta V_{\text{out}}\). Because \(V_{\text{out}}=G(V_+-(V_{-}-\beta V_{\text{out}}))=A(V_+-V_-)\), it follows that the closed-loop gain \(G\) is related to the open-loop gain \(A\) via:

\[G=\frac{A}{1+\beta A}\]

In particular, because \(A\gg 1\) is large, \(G\approx 1/\beta\) is essentially independent of \(A\)! And even if there’s a large manufacturing uncertainty \(\Delta A\) in the open-loop gain \(A\), the corresponding uncertainty \(\Delta G=\frac{\Delta A}{(1+\beta A)^2}\) in the closed-loop gain \(G\) (assuming \(\Delta\beta\ll\Delta A\) has much less uncertainty which is true) is utterly negligible due to the suppression by a much larger factor of \((1+\beta A)^2\) in the denominator. Intuitively, this is just saying that all the way out at \(A\approx 10^5\), the blue curve is basically flat at \(G\approx 1/\beta\) and so moving around a little bit \(\sim\Delta A\) out so far on such a flat curve won’t really affect the fact that \(G\) is still \(\approx 1/\beta\).

Finally, there is a whole pantheon of clever op-amp circuits that have been devised such as active filters, differentiators, integrators (to make it practical/avoid the DC railing problem, add a shunt resistor in parallel with the capacitor!), summing amplifiers, differential amplifiers, instrumentation amplifiers, comparators, precision rectifiers, \(IV\) converters, Schmitt triggers, etc.

One more point regarding op-amps is that, for AC voltage inputs, their (now complex) open-loop gain \(A=A(\omega)\in\textbf C\) is a function of frequency \(\omega\in\textbf R\), roughly going like \(|A(\omega)|\sim\omega^{-1}\) as \(\omega\to\infty\). On a Bode magnitude plot of \(\log|A(\omega)|\) vs. \(\log\omega\), this behavior shows up as an asymptotic \(\omega\to\infty\) linear behavior with slope \(\frac{d\log|A(\omega)|}{d\log\omega}=-1\). Op-amps are designed with this kind of low-pass open-loop behavior to avoid instabilities at large \(\omega\). In addition to this, one can also add decoupling capacitors \(C_{\text{decoupling}}\) between the external power supply leads \(\pm V_S\) and ground \(\text{GND}\) to short out high-frequency noise that occurs such as when turning on the external DC power supply on (the Heaviside step function \(V_0[t>0]\) has Fourier transform \(V_0\left(\frac{1}{i\omega\sqrt{2\pi}}+\sqrt{\frac{\pi}{2}}\delta(\omega)\right)\)). Finally, one more technicality is that in real life op-amps are also limited by a slew rate \(\dot V_{\text{out}}^*\) such that for all \(t\in\textbf R\), \(|\dot V_{\text{out}}(t)|<\dot V_{\text{out}}^*\) is capped by the op-amp’s slew rate.

Control Systems

A control system may be thought of abstractly as a dynamical system (labelled as just “system” in the block diagrams below) subject to an additional controller input \(\textbf u\) as follows:

\[\dot{\textbf x}=\textbf F(\textbf x,\textbf u)\]

Generalizing the situation for op-amps, there are two kinds of control systems, namely open-loop (also called feedforward) control systems and closed-loop (also called feedback) control systems:

Note that even in the so-called “open-loop control system” there is a closed loop from the output state \(\textbf x\) back into the system. This is just the fact that we’re working with a dynamical system \(\dot{\textbf x}=\textbf F(\textbf x,…)\) where the state at a later time depends on the state at a previous time (we can of course also consider the special case where \(\partial\textbf F/\partial\textbf x=0\) for the system of interest so that there wouldn’t be any such closed loop). This same closed loop also appears in the closed-loop control system, but here the novelty is that there is an additional closed loop (called the control loop!) from the output \(\textbf x\) all the way back into the controller (in the form of the error \(\textbf x-\textbf x_0\) from the desired reference setpoint \(\textbf x_0\)). It is usually this kind of control system possessing a closed control loop in a negative feedback configuration (seeking to minimize \(\textbf x-\textbf x_0\)) that tends to be the most useful. Practically speaking, this is implemented by adding a sensor to continuously the monitor the system’s state \(\textbf x\) and having it feed the measured state \(\textbf x\) back into the controller.

To give an example, one of my pet peeves is traffic lights. Even when there are no cars anywhere in the transverse lane, one must typically still wait out the entire duration of the red light turning green before one can go. In the language introduced above, this is an example of an open-loop control system; the dynamical system consists of the cars approaching this intersection (with state vector \(\textbf x:=(N_1,N_2)\) the number of cars lined up at each intersection) and the controller is the traffic lights. It sucks. To improve this control system, it makes to sense to add a negative feedback control loop (e.g. cameras for counting the number of cars \(N_1,N_2\)). The setpoint state would then be \((N_1,N_2)=(0,0)\).

Focusing on the more interesting case of closed-loop control systems with a negative feedback control loop, one can view the control system in \(\omega\)-space (or engineers also like \(s=i\omega\)-space) rather than the \(t\)-domain. If one assumes that the dynamical system is linear, then one has a block diagram of the form:

where now \(H_C(\omega)\) is the transfer function of the controller and \(H_S(\omega)\) the transfer function of the dynamical system. Here, because \(\hat{\textbf u}=H_C(\hat{\textbf x}-\hat{\textbf x}_0)\) and \(\hat{\textbf x}=H_S\hat{\textbf u}\), one can eliminate the controller input \(\hat{\textbf u}\) to obtain:

\[\hat{\textbf x}=(H_L-1)^{-1}H_L\hat{\textbf x}_0\]

where \(H_L=H_SH_C\) is the transfer function for the entire control loop.

Whenever \(|H_L|>1\) and \(\text{arg}(H_L)=\pi\), one gets a subcritical Hopf bifurcation (intuitively makes sense but exact details are still muddled). This is like saying that the negative feedback inadvertently becomes positive feedback becomes the application of the negative feedback is \(\pi\)-out of phase with the system’s dynamics, accidentally acting like positive feedback instead. To quantify how far a given (closed-loop, negative-feedback) control system is from this oscillatory instability, one introduces the two complementary notions of the gain margin \(||H_L|-1|_{\arg(H_L)=\pi}\) and phase margin \(|\arg(H_L)-\pi|_{|H_L|=1}\) which are typically straightforward to read off from Bode magnitude and phase plots of the control loop transfer function \(H_L\).

One very popular and effective type of controller is a PID (proportional, integral, derivative) controller which operates on the following formula:

\[\textbf u_{\text{PID}}=K_P\Delta\textbf x+K_I\int_0^t\Delta\textbf x(t’)dt’+K_D\Delta\dot{\textbf x}\]

where \(\Delta\textbf x:=\textbf x-\textbf x_0\) and \(K_P,K_I,K_D\) are constants that typically need to be tuned via trial-and-error.

Noise Filtering

In general, noise \(N(t)\) in the \(t\)-domain may be construed as an uncountable continuum of draws from an uncountable continuum of continuous random variables \(\mathcal N(t)\) that could in principle themselves be \(t\)-dependent (and possibly correlated across different times \(\mathcal N(t)\sim\mathcal N(t’)\)), one draw and one random variable per moment \(t\in\textbf R\) in time (this is often called a stochastic process). Of course, one can also opt to view the noise \(N(t)\mapsto\hat N(\omega)\) in \(\omega\)-space. In this case one often defines \(|\hat N(\omega)|^2\) to be the energy spectral density of the noise \(N\), and because the total energy \(E:=\int_{-\infty}^{\infty}|\hat N(\omega)|^2d\omega=\int_{-\infty}^{\infty}|N(t)|^2dt\) tends to diverge \(E=\infty\) for these kinds of noisy signals \(N\), it is instead more instructive to look at the total power \(P\) of the noise \(N\) instead \(P:=\lim_{\Delta t\to\infty}\frac{1}{\Delta t}\int_{-\Delta t/2}^{\Delta t/2}|N(t)|^2dt=\lim_{\Delta t\to\infty}\frac{1}{\Delta t}\int_{-\infty}^{\infty}|N(t)\text{rect}(t/\Delta t)|^2dt=\lim_{\Delta t\to\infty}\frac{1}{\Delta t}\int_{-\infty}^{\infty}|\hat N(\omega)*\text{sinc}(\omega\Delta t)|^2d\omega\). In particular, the integrand then defines the power spectral density \(\hat P(\omega)=\lim_{\Delta t\to\infty}\frac{1}{\Delta t}|\hat N(\omega)*\text{sinc}(\omega\Delta t)|^2\). Defining the autocorrelation of the noise as \(A(t):=\int_{-\infty}^{\infty}N(t’)N(t’-t)dt’\), a quick calculation via the convolution theorem (assuming a stationary random/Wiener stochastic process which is valid for noise \(N\)) allows one to check that:

\[\hat P(\omega)=\hat A(\omega)\]

The fact that the autocorrelation and the power spectral density are Fourier transform pairs is called the Wiener-Knichin theorem.

There are several different kinds of noise commonly encountered. The simplest is called white noise. This is defined by having a uniform power spectral density \(\hat P_{\text{white}}(\omega)=\text{constant}\). In this case, the Wiener-Knichin theorem asserts that the autocorrelation of white noise is a Dirac delta distribution \(A_{\text{white}}(t)\sim \delta(t)\). This means the only way you get a non-zero correlation is when you put the noise exactly on top of itself; this means that white noise is uncorrelated noise. Examples of white noise include Johnson-Nyquist noise \(\hat P_{\text{JN}}(\omega)\propto T\) in electronics (manage by cooling down electronics) and shot noise in photodiodes.

There is also pink noise which is defined by having a power spectral density which (no pun intended) goes as the power law \(\hat P_{\text{pink}}(\omega)\sim\frac{1}{\omega}\). Roughly speaking, pink noise is universal because it’s associated with self-similarity, similar to a fractal. One simple way to see this self-similarity is that the power allocated to each “octave” (or generalization thereof) is constant for instance:

\[\int_{\omega_1}^{\lambda\omega_1}\hat P_{\text{pink}}(\omega)d\omega=\int_{\omega_2}^{\lambda\omega_2}\hat P_{\text{pink}}(\omega)d\omega\]

for all \(\lambda,\omega_1,\omega_2\in\textbf R\) by virtue of how logarithms work (the autocorrelation for pink noise is also \(A(t)\sim 1/|t|\) if one interprets its power spectral density as \(\hat P_{\text{pink}}(\omega)\sim 1/\omega=1/|\omega|\) symmetrically). Finally, there are also kinds of noise like \(\hat P_{\text{Brownian}}(\omega)\sim 1/\omega^2\).

In practice, it is common to have both white noise and pink noise mixed in with a signal. In the DC limit \(\omega\to 0\), it is clear that pink noise will dominate over white noise \(\hat P_{\text{pink}}(\omega)\gg \hat P_{\text{white}}(\omega)\) but in the high-frequency limit it is white noise that will dominate \(\hat P_{\text{white}}(\omega)\gg \hat P_{\text{pink}}(\omega)\).

Lock-In Detection

A common situation one encounters is that the signal \(f(t)\) one is interested might have some frequency component at some small \(\omega_0\in\textbf R\) so that by the above discussion it will be buried in pink noise. In that case, provided one knows a priori what the frequency \(\omega_0\) of the signal is that one is interested in, then the technique of lock-in detection allows one to lock on to the signal at that frequency \(\omega_0\) notwithstanding even an \(\text{SNR}<1\) (i.e. being buried in the pink noise)! Roughly speaking, lock-in detection just amounts to implementing the Fourier transform \(\hat f(\omega_0)\) evaluated at the frequency \(\omega=\omega_0\) of interest. One starts by having some reference oscillator generate a relatively clean sinusoidal signal \(\sim\cos(\omega_0 t)\) at the frequency \(\omega_0\) of interest. Then, a “mixer” multiplies (or “modulates”) the noisy signal \(f(t)\) by this clean sinusoid (or sometimes a square wave) to obtain \(\sim f(t)\cos(\omega_0 t)\). Finally one needs some kind of low-pass filter to implement an integral-like step \(\sim\int f(t)\cos(\omega_0 t)dt\), which is the usual Fourier transform for \(f\) evaluated at frequency \(\omega_0\) (strictly one has to first also mix in a \(\sim\sin(\omega_0 t)\) reference signal too and repeat everything above). Because the Fourier transform of course not only provides amplitude information \(|\hat f(\omega_0)|\) but also phase information \(\text{arg}\hat f(\omega_0)\), lock-in detection is also sometimes called “phase-sensitive detection”.

Problem: A resistor \(R\) sitting on a table, connected to nothing whatsoever will of course have \(V=0\) across it…or to be more precise, the expected voltage \(\langle V\rangle=0\) will be vanishing, but it turns there will be thermal/equilibrium fluctuations \(\langle V^2\rangle\neq 0\) about \(\langle V\rangle=0\).

Explain heuristically what the origin of such voltage fluctuations \(\langle V^2\rangle\neq 0\) (also called Johnson noise) is and obtain a quantitative estimate of their order of magnitude by equating, in equilibrium, the power dissipated in the resistor with its thermal power from blackbody radiation in the Rayleigh-Jeans limit.

Solution: One has:

\[\frac{\langle V^2\rangle}{(2R)^2}R\sim k_BT\Delta f\]

Hence the RMS voltage is:

\[\sqrt{\langle V^2\rangle}=\sqrt{4Rk_BT\Delta f}\]

where the factor of \(4\) comes from a more detailed analysis (e.g. see Nyquist’s \(1928\) paper about this). In fact, Johnson noise is just a corollary of the fluctuation-dissipation theorem.

Problem #\(1\):

Solution #\(1\):

Problem:

Solution:

Problem #\(3\):

Solution #\(3\):

Problem #\(4\):

Solution #\(4\):

Problem #\(5\):

Solution #\(5\): First, although this tight-binding model looks like a classical model, in fact it arises from the quantum Hamiltonian \(H=E_01-t_1\sum_n(|n+1\rangle\langle n|+|n-1\rangle\langle n|)\) together with the “discrete position representation” \(|\psi\rangle=\sum_n\psi_n|n\rangle\Leftrightarrow\psi_n:=\langle n|\psi\rangle\). The piece \(\psi_{n-1}+\psi_{n+1}\) can be coarse-grained to a kinetic energy:

\[\psi_{n-1}+\psi_{n+1}=\psi_{n-1}+\psi_{n+1}-2\psi_n+2\psi_n\approx\Delta x^2\psi^{\prime\prime}-2\psi\]

to yield the Schrodinger equation for a free particle:

\[-t_1\Delta x^2\psi^{\prime\prime}+(E_0-2t_1)\psi=E\psi\]

with effective mass \(m^*\) defined through \(\frac{\hbar^2}{2m^*}=t_1\Delta x^2\). The eigenstates are the usual scattering plane waves \(\psi(x)\sim e^{\pm ikx}\) where one has the free particle dispersion relation:

\[E=E_0-2t_1+t_1k^2\Delta x^2\]

This motivates how to proceed, namely replace \(x\mapsto n\Delta x\) and make the ansatz \(\psi_n=e^{ikn\Delta x}\). Doing so gives the cosine band dispersion:

\[E=E_0-2t_1\cos k\Delta x\]

so in this tight-binding approximation there is a single band of width \(4t_1\) centered at \(E_0\) (no notion of band gaps in this model because there’s only \(1\) band!). Notice also that Taylor expanding the \(\cos\) reproduces the earlier free particle dispersion near \(k=0\).

For part b), for \(n\leq -1\) or \(n\geq 2\), in both cases one finds:

\[t_1(c+c^{-1})+E-E_0=0\]

Meanwhile for \(n=0,1\), one finds:

\[\begin{pmatrix}s&E-E_0+ct_1\\E-E_0+ct_1&s\end{pmatrix}\begin{pmatrix}\alpha\\\beta\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\]

this is a \(2\times 2\) circulant matrix so the (unnormalized) eigenvectors \((\alpha,\beta)=(1,\pm 1)\) are very easy to read off and so no determinant computations are even needed; the dispersion relations are \(E-E_0=-ct_1\pm s\). Eliminating \(E-E_0\) in the “bulk” equation immediately yields \(c=\pm t_1/s\) so that \(|c|<1\) which indicates a bound state in which the electron is localized near the origin by the high-energy \(s>t_1\) defect because \(\psi_n\to 0\) as \(n\to\pm\infty\). The energies \(E_{\pm}=E_0\pm\left(\frac{t_1^2}{s}+s\right)\) may be checked to lie outside the original band \(E\in[E_0-2t_1,E_0+2t_1]\).

Problem #\(6\):

Solution #\(6\):

Problem #\(7\):

Solution #\(7\):

Problem #\(1\): Derive the Maxwell relation for a gas:

\[\left(\frac{\partial S}{\partial V}\right)_T=\left(\frac{\partial p}{\partial T}\right)_V\]

And explain why Maxwell relations in general should be viewed as much more than just mathematical identities.

Solution #\(1\): Here it is clear that one is viewing \((T,V)\) as independent variables, and the corresponding energy which has these as its natural variables is the Helmholtz energy \(F\). This has the differential:

\[dF=-SdT-pdV\]

So by Clairaut’s theorem:

\[\frac{\partial^2F}{\partial T\partial V}=\frac{\partial^2F}{\partial V\partial T}\]

\[\left(\frac{\partial p}{\partial T}\right)_V=\left(\frac{\partial S}{\partial V}\right)_T\]

All Maxwell relations (for a gas system described by state space \((p,V)\)) can be boiled down to the Jacobian determinant:

\[\frac{\partial (p,V)}{\partial (T,S)}=1\]

which expresses the area/orientation-preserving property of the state space transformation \((p,V)\to(T,S)\). So the point is that thermodynamics is really a theory of geometry (cf. special relativity), the geometry of equilibrium states.

In any Maxwell relation, one partial derivative represents a process which is experimentally accessible, while the other side represents an experimentally difficult quantity to measure (this is typically the entropy \(S\)) but which is in some way seeking to understand microscopic detail. Even to measure (changes in) entropy, one typically measures heat capacities…but the partial derivative is being taken at constant \(T\)! So always write Maxwell relations in a way that agrees with the usual convention in math of putting the dependent variable on the LHS and independent variables on the RHS, in this case putting the experimentally difficult partial derivative on the LHS and the experimentally accessible one on the RHS.

Problem #\(2\): For a gas, what does it mean to have complete thermodynamic information?

Solution #\(2\): Knowing \(N,V,T,\mu,p,S\). For instance, if one specifies a function \(F=F(N,V,T)\) for the Helmholtz energy, then this provides one with complete thermodynamic information since one can obtain \(\mu=\frac{\partial F}{\partial N}, p=-\frac{\partial F}{\partial V},S=-\frac{\partial F}{\partial T}\). Since the Helmholtz free energy is equivalent to giving the canonical partition function \(F=-k_BT\ln Z\), this is unsurprising.

Problem #\(3\): What are the assumptions required for each of the following differential equalities to be true?

\[dE=\bar dQ+\bar dW\]

\[\bar dQ=TdS\]

\[\bar dW=-pdV+\mu dN\]

\[dE=TdS-pdV+\mu dN\]

Solution #\(3\): All \(4\) equations require one to go between \(2\) (infinitesimally separated) equilibrium states. In other words, this is thermodynamics! With this in mind, the first equation is then always true regardless of the path one takes between those \(2\) equilibrium states (because \(E\) is a state function). However, the \(2^{\text{nd}}\) and \(3^{\text{rd}}\) equations are true iff the path by which one goes between those \(2\) equilibrium states is reversible (since \(\bar dW\) and \(\bar dQ\) are now path functions); if the path were irreversible then both equalities would instead become inequalities! Finally, although at first one may think the \(4^{\text{th}}\) equation is only true for a reversible path, in fact because \(dE\) is an exact \(1\)-form (whereas \(\bar dW,\bar dQ\) are inexact), it is again always true, reversible or irreversible, since at the end of the day all that matters for state functions is the initial and final states.

In the case of \(E\) for a gas, clearly its extensive natural variables are \(N,V,S\), while its intensive derived variables are \(\mu, p, T\).

This also makes it clearer what the \(1^{\text{st}}\) law of thermodynamics is really saying, i.e. the sum of the inexact \(1\)-forms \(\bar dQ+\bar dW\) is (nontrivially!) an exact \(1\)-form that one happens to call \(dE\). Put another way, it is rooted in a bedrock belief that conservation of energy should hold even in the presence of heat transfer \(\bar dQ\).

Problem #\(4\): What is the difference between a quasistatic and a reversible process?

Solution #\(4\): Reversible processes are a subset of quasistatic processes. More precisely:

\[\text{Reversible}=\text{Quasistatic}+\text{No Friction}\]

where \(\text{No Friction}\) can also be interpreted as “adiabatic” or “isentropic”. Reversible really means time reversible.

Problem #\(5\): Prove Euler’s homogeneous function theorem, i.e.:

\[V(\lambda\textbf x)=\lambda^n V(\textbf x)\Leftrightarrow \textbf x\cdot\frac{\partial V}{\partial\textbf x}=nV(\textbf x)\]

Hence, by appealing to extensivity of the energy \(E=E(N,V,S)\), obtain the Gibbs-Duhem relation:

\[SdT-Vdp+Nd\mu=0\]

Solution #\(5\): Just differentiate \(\left(\frac{\partial}{\partial\lambda}\right)_{\lambda=1}\) (recognize this as the precursor of the virial theorem!). Extensivity of the energy is equivalent to the \(n=1\) case of Euler’s homogeneous function theorem, i.e.

\[N\frac{\partial E}{\partial N}+V\frac{\partial E}{\partial V}+S\frac{\partial E}{\partial S}=E\]

\[E=TS-pV+\mu N\]

Taking the differential of both sides:

\[dE=TdS-pdV+\mu dN+SdT-Vdp+Nd\mu\]

so because \(dE=TdS-pdV+\mu dN\), the latter part must vanish, which is the Gibbs-Duhem relation.

Problem #\(6\): Using the Gibbs-Duhem relation, prove the Clausius-Clapeyron equation for a single-component system at a \(1^{\text{st}}\)-order phase boundary/coexistence curve:

\[\frac{\partial p}{\partial T}=\frac{\Delta S}{\Delta V}\]

Solution #\(6\): For each phase of the single-component system, one has a Gibbs-Duhem relation which can be written in the form:

\[S_1dT_1-V_1dp_1+N_1d\mu_1=0\]

\[S_2dT_2-V_2dp_2+N_2d\mu_2=0\]

Now, the defining property of a phase boundary is that the \(2\) phases are in equilibrium, so the intensive variables between the \(2\) phases are equal everywhere on the phase boundary \(T_1=T_2:=T,p_1=p_2:=p,\mu_1=\mu_2:=\mu\). So moving infinitesimally along this phase boundary, it is a rigorous corollary that \(dT_1=dT_2:=dT,dp_1=dp_2:=dp,d\mu_1=d\mu_2:=d\mu\). But now one has \(2\) equations and \(3\) unknowns \(dT,dp,d\mu\):

\[S_1dT-V_1dp+N_1d\mu=0\]

\[S_2dT-V_2dp+N_2d\mu=0\]

Eliminating \(d\mu\) yields the Clausius-Clapeyron equation:

\[\frac{dp}{dT}=\frac{S_2/N_2-S_1/N_1}{V_2/N_2-V_1/N_1}\]

Or in terms of molar entropies \(s:=S/n\) and molar volumes \(v:=V/n\), where \(n:=N/N_A\):

\[\frac{dp}{dT}=\frac{\Delta s}{\Delta v}\]

So in other words, phase transitions are all about parameters changing discontinuously, and in this case the discontinuous changes \(\Delta s,\Delta v\) are directly also what influence the slope of the phase boundary.

Because the phase transition occurs isothermally at temperature \(T\), one can also write \(\Delta s:=q_L/T\) where \(q_L\) is the molar latent heat released during the phase transition. And often, if it’s a liquid-to-gas phase transition, the molar volume of liquid water \(v_{\ell}\approx\) is significantly less than the molar volume of water vapor \(v_{\text H_2\text O(\text g)}\approx\), so it is common to approximate \(\Delta v\approx v_g\approx RT/p\) (if the gas is ideal) so that \(q_L\) refers to the specific latent heat of vaporization \(\ell\to\text g\) and, assuming \(q_L\) to be approximately \(T\)-independent, the Clausius-Clapeyron equation can be integrated to yield the equation of the phase boundary itself:

\[p=p_0\exp-\frac{q_L}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\]

where in this context \(p\) is called the vapor pressure. Alternatively, written in the form:

\[\ln p-\ln p_0=-\frac{q_L}{R}\left(\frac{1}{T}-\frac{1}{T_0}\right)\]

shows that \(q_L\) can be experimentally determined through linear regression of \(\ln p\) vs. \(1/T\).

Finally, note that earlier the choice was made to eliminate \(d\mu\) to isolate for \(dp/dT\); however one could just as well have eliminated \(dp\) to isolate for \(d\mu/dT\) or eliminated \(dT\) to obtain \(dp/d\mu\)…each giving rise to its own kind of Clausius-Clapeyron equation.

Problem #\(7\): Using the fact that in a closed cycle \(\oint dE=0\), write \(dE=TdS-pdV+\mu dN\) and apply Stokes’ theorem to obtain suitable Maxwell relations.

Solution #\(7\): Stokes’ theorem needs a curve and a surface with that as its boundary curve. First, consider looking at just motion in the \((S,V)\)-plane so that \(dN=0\). Then Stokes’ theorem reduces to Green’s theorem in that plane:

\[0=\oint TdS-pdV=\oint\begin{pmatrix} T\\-p\end{pmatrix}\cdot\begin{pmatrix}dS\\dV\end{pmatrix}=\iint\left(-\frac{\partial p}{\partial S}-\frac{\partial T}{\partial V}\right)dSdV\]

Hence one obtains the Maxwell relation:

\[\left(\frac{\partial p}{\partial S}\right)_{N,V}=-\left(\frac{\partial T}{\partial V}\right)_{N,S}\]

Working in the \((N,S)\) plane or \((N,V)\) plane produces \(2\) other Maxwell relations.

Problem #\(8\): Equation of state as a constitutive relation/dispersion relation/heart of the physics. Want to distinguish clearly between this and all the kinematic Maxwell relations, definitions, etc.

Equations of state will never involve entropy \(S\), the experimentally inaccessible bastard. So when one encounters any \(\partial S\) quantities, the immediate knee-jerk reaction should be to convert it to a corresponding \(\partial T\) derivative which is more readily measurable.

All the usual shorthands for special collections of partial derivatives like heat capacities, thermal expansion coefficients, compressibilities/bulk moduli, and other moduli are singled out for this special treatment because they are often found to be constant material parameters?

\[C_V:=\left(\frac{\partial E}{\partial T}\right)_V\]

\[C_{\sigma}:=\]

\[\alpha=\left(\frac{\partial V}{\partial T}\right)_{\sigma}\]

\[\kappa_T=\frac{1}{V}\left(\frac{\partial V}{\partial\sigma}\right)_T\]

Solution #\(8\):

Problem: In general, the energy may be written abstractly as a linear combination of \(N\) extensive natural variables \(Q_i\) (thought of as “generalized charges/coordinates“) weighted by their \(N\) conjugate intensive derived variables \(\phi_i\) (thought of as “generalized potentials/forces“):

\[E=\sum_{i=1}^N\phi_iQ_i\]

For instance, temperature \(T\) can be thought of as an “entropy potential” as entropy \(S\) flows via heat from high \(T\) to low \(T\). Similarly, \(\sigma=-p\) is a “volume potential”, thus volume \(V\) flows from high \(\sigma\) to low \(\sigma\), aka from low \(p\) to high \(p\) (this is consistent with the usual interpretation of pressure \(p\) as a force acting on the piston walls). Similarly, particles \(N\) flow from high chemical potential \(\mu\) to low \(\mu\). In general, charge \(Q_i\) flows from high potential \(\phi_i\) to low \(\phi_i\).

How many Maxwell relations can be obtained from \(E\) alone and what is their general form?

Solution: There are \(N\choose{2}\)\(=\frac{N(N-1)}{2}\) Maxwell relations that can be wringed out from this equilibrium potential \(E\) alone. Because there are no fiddly minus signs in this series, it is clear that there won’t be any fiddly minus signs in the corresponding Maxwell relations. In this case, all Maxwell relations will have the form:

\[\left(\frac{\partial\phi_1}{\partial Q_2}\right)_{Q_1,Q_3,…}=\left(\frac{\partial\phi_2}{\partial Q_1}\right)_{Q_2,Q_3,…}\]

Problem: For a closed single-component \(3\)D gas in equilibrium, how many independent intensive variables parameterize the equilibrium manifold of the system?

Solution: The Gibbs phase rule (analogous to Euler’s graph formula \(F+V=E+2\)) asserts:

\[I+P=C+2\Rightarrow I+1=1+2\Rightarrow I=2\]

So one is always free to select any \(2\) intensive potentials such that when their values are fixed, so too automatically are the values of all other intensive potentials at equilibrium. It’s a bit like saying if a mass is acted on by forces \(\textbf F_1,\textbf F_2,\textbf F_3\), and one is told \(\textbf F_1,\textbf F_2\), then because the mass is in translational equilibrium the value of \(\textbf F_3=-\textbf F_1-\textbf F_2\) is fixed.

Problem: For a single-component \(3\)D gas with energy/Hamiltonian:

\[E=TS+\sigma V+\mu N\]

Explain whether the following partial derivatives are (in general) well-posed or not. For those that are well-posed, write down their associated Maxwell relation.

\[\left(\frac{\partial S}{\partial V}\right)_{N}\]

\[\left(\frac{\partial S}{\partial V}\right)_{T,V,\mu}\]

\[\left(\frac{\partial S}{\partial V}\right)_{T,N}\]

\[\left(\frac{\partial T}{\partial V}\right)_{\sigma, N}\]

\[\left(\frac{\partial T}{\partial \mu}\right)_{\sigma, N}\]

\[\left(\frac{\partial T}{\partial S}\right)_{\sigma, N}\]

\[\left(\frac{\partial T}{\partial E}\right)_{V, N}\]

\[\left(\frac{\partial H}{\partial T}\right)_{V, N}\]

\[\left(\frac{\partial T}{\partial G}\right)_{\sigma, \mu}\]

\[\left(\frac{\partial V}{\partial \mu}\right)_{S,E}\]

\[\left(\frac{\partial T}{\partial V}\right)_{S,\sigma}\]

Solution: It is useful to define the notion of a natural variable set to be any energy together with its set of natural variables which are the natural variables of some energy. Starting from the fundamental natural variable set:

\[\{E,S,V,N\}\]

Legendre transforms give all \(8\) of the other natural variable sets:

\[\{F,T,V,N\}\]

\[\{H,S,\sigma,N\}\]

\[\{G,T,\sigma,N\}\]

\[\{\Phi,T,V,\mu\}\]

and \(3\) other combinations that don’t seem to have a name. Anyways, the point is that the Gibbs phase rule gives \(I=2\), but it doesn’t count the extensivity degree of freedom which is always present because that doesn’t affect equilibrium, hence explaining why all natural variable sets have \(2+1=3\) variables. Moreover, as should be clear from the fact that one is Legendre transforming, no variable appears with its conjugate in the same natural variable set.

This then provides a litmus test for whether a partial derivative is ill-posed or not; just see if the variables in the subscript together with either the variable in the numerator or the variable in the denominator can be made to form a natural variable set or not; more simply, can one form a set which is not cohabited by conjugate variables (is there a notion of conjugate variables for the energies like \(E,F,\)etc, themselves that makes the Maxwell relations continue to hold)?

(aside: actually this last partial derivative should be undefined? because holding both \(\sigma,\mu\) constant amounts to holding \(T\) constant since they’re intensive…?)

Also, given any \(2\) extensives \(E_1,E_2\) and any \(2\) distinct intensives \(I_1\neq I_2\), the partial derivative:

\[\left(\frac{\partial E_1}{\partial E_2}\right)_{I_1,I_2}=\frac{E_1}{E_2}\]

this follows because \(E_1/E_2\) is intensive so equal to a constant \(\lambda=\lambda(I_1,I_2)\), so \(E_1=\lambda E_2\Rightarrow \partial E_1/\partial E_2=\lambda=E_1/E_2\).

Aside: when intensive and intensive or extensive and extensive bunch together like bosons, then the Maxwell relation has a minus sign. Similarly, intensive and extensive antibunch like fermions but that gives a + sign Maxwell relation. It seems that, roughly speaking, the individual intensive/extensive variables can treated like fermions, in the sense that starting with any Maxwell relation and exchanging \(\phi_i\Leftrightarrow Q_i\) gives a minus sign. And obviously equality is symmetric which is really a reflection of the fact that 2 fermions together make a boson.

Problem: For a single-component system, what are the \(3\) standard intensive equilibrium material properties?

Solution: The compressibility (either the isothermal one \(\kappa_T\) or the isentropic one \(\kappa_S\), and equivalently one can use the isothermal bulk modulus \(B_T\) or the isentropic bulk modulus \(B_S\)), the specific heat capacity (either \(c_V\) or \(c_p\) and note it could be per unit mass or per unit mole, etc.) and the thermal expansion coefficient \(\alpha\). The definitions are here.

Problem: Using the compressible Bernoulli’s equation, show that enthalpy is conserved in a Joule-Thomson expansion. Define the corresponding Joule-Thomson coefficient \(\mu_{\text{JT}}\) and show that \(\mu_{\text{JT}}=0\) for ordinary Joule expansion of an ideal gas.

Solution: The compressible Bernoulli equation looks like the usual Bernoulli equation but with the addition of the gas’s specific energy \(e\):

\[p+\frac{1}{2}\rho v^2+\rho\phi+e=\text{const.}\]

The terms \(h:=e+p\) are nothing more than the specific enthalpy of the gas.

In Joule-Thomson expansion, it is conventional to assume the macroscopic energy density \(\frac{1}{2}\rho v^2+\rho\phi\) is constant throughout the expansion, so this implies that \(h\) is conserved.

Alternatively, one can imagine a setup in which gas at a higher (but constant) pressure \(p\) is throttled through a porous plug to a region of lower (constant) pressure \(p'<p\). Then, the gas behind does work \(pV\) on the gas that passes through, and similarly the gas that expands in the other side does work \(p’V\) on the gas in front of it (too lazy to draw this). It’s as if there were fictitious pistons on either side of the plug…assuming the whole thing is enclosed in adiabatic walls, so \(Q=0\), then from the first law of thermodynamics, \(\Delta H=0\).

The Joule-Thomson temperature change \(\Delta T_{\text{JT}}\) is familiar in everyday life. For instance, when opening a bike tire valve, the pressure inside is initially a few atmospheres higher than the ambient atmospheric pressure, but as the gas escapes (isenthalpically) it cools down, causing the tire valve to feel cold to the touch. Physically, for a non-ideal gas, expansion increases potential energy, reducing kinetic energy, hence cooling the gas.

In general, the Joule-Thomson coefficient:

\[\mu_{\text{JT}}:=\left(\frac{\partial T}{\partial p}\right)_{H,N}=\frac{V}{C_p}\left(\alpha T-1\right)\]

quantifies this cooling across a pressure differential (for most gases at room temperature, the Joule-Thomson coefficient is positive and of order \(\mu_{\text{JT}}\sim 0.1\frac{\text J}{\text{atm}}\), hence explaining the cooling rather than heating normally observed).

For an ideal gas, \(\alpha=1/T\) so there is no associated Joule-Thomson cooling/heating across a pressure differential. The inversion point is when \(\mu_{\text{JT}}=0\).

Problem: Distinguish between heat engines, refrigerators, and heat pumps.

Solution: A heat engine \(\textbf x_E(t)\) is an abstraction of any periodic process \(\textbf x_E(t+\Delta t)=\textbf x_E(t)\) whose net outcome in each period \(\Delta t\) is to convert some amount of (useless!) heat \(Q_H>0\) into (useful!) work \(W>0\) (thus, both \(Q_H\) and \(W\) are normalized per orbit of the heat engine \(\textbf x_E(t)\) in a suitable state space). It thus makes sense to define the efficiency \(\eta\) of a heat engine by the buck-to-bang ratio:

\[\eta:=\frac{W}{Q_H}\]

Kelvin’s formulation of the \(2^{\text{nd}}\) law of thermodynamics is logically equivalent to the assertion that the efficiency \(\eta\) of any heat engine \(\textbf x_E(t)\) must obey:

\[\eta<1\]

though in theory \(\eta\) can get arbitrarily close to \(1\). Schematically, for each orbit:

Since \(E\) is a function of the engine’s state \(\textbf x_E(t)\) only, over each \(\Delta t\)-orbit, \(\oint dE=0\) so the first law of thermodynamics guarantees \(\oint\bar dQ+\oint\bar dW=0\). This guarantees that \(Q_H=W+Q_C\) over each engine period \(\Delta t\).

Both refrigerators and heat pumps are basically the same thing (note: refrigerators are not called “heat refrigerators” even though that would have been more consistent with “heat engine” and “heat pump”). They are also both abstractions of any periodic process whose net outcome in each period is to remove some amount of heat \(Q_C>0\) from a colder place and dump some of that heat \(Q_H>0\) into a hotter place, fighting an uphill battle against the spontaneous direction that heat would otherwise flow (i.e. from hotter to colder). The difference between refrigerators and heat pumps is simply a matter of emphasis; in the case of refrigerators, the goal is to remove as much heat \(Q_C\) from the cold place as possible, whereas for heat pumps the goal is to dump as much heat \(Q_H\) into the hot place as possible. Schematically, for each orbit:

Clausius’s formulation of the \(2^{\text{nd}}\) law of thermodynamics is logically equivalent to the assertion that for any refrigerator or heat pump, \(Q_H<Q_C\). This implies that some external work \(W>0\) must be done each cycle to facilitate this heat transfer. This motivates the corresponding definitions of the coefficient of performance \(\text{COP}\) for a refrigerator:

\[\text{COP}:=\frac{Q_C}{W}\]

and a heat pump:

\[\text{COP}:=\frac{Q_H}{W}\]

Problem: Explain what the reversibility theorem (also misleadingly called “Carnot’s theorem”) asserts.

Solution: The notions of heat engines and refrigerators/heat pumps are very general, and a priori there is no requirement about “using a hot reservoir \(T_H\) and a cold reservoir \(T_C\)”. However, if one restricts one’s scope to just the subset of heat engines and refrigerators/heat pumps operating only between \(2\) heat reservoirs \(T_H,T_C\), then within this subset one can prove that reversible cycles are the most efficient in all cases (i.e. for heat engines they maximize \(\eta\), and for refrigerators/heat pumps they maximize \(\text{COP}\)).

Problem: Show that the specific example of a Carnot cycle is reversible (though it certainly isn’t the only reversible cycle one can take), and hence compute \(\eta\) for a Carnot heat engine and \(\text{COP}\) for both a Carnot refrigerator and a Carnot heat pump, thus obtaining upper bounds on these values within the subsets described earlier.

Solution: The key is that every step of the Carnot cycle is reversible, regardless of whether it is run as a heat engine or as a refrigerator/heat pump. The universal way to depict the Carnot cycle is on the \((T,S)\)-plane:

Any other depiction of the Carnot cycle, such as in the \((p,V)\)-plane using an ideal gas working substance is then simply a geometric deformation of this rectangle:

For a Carnot heat engine, the efficiency is:

\[\eta=1-\frac{T_C}{T_H}<1\]

For a Carnot refrigerator, the coefficient of performance is:

\[\text{COP}=\frac{T_C}{T_H-T_C}\]

And for a Carnot heat pump:

\[\text{COP}=\frac{T_H}{T_H-T_C}\]

Problem: By a Stokes-like maneuver, any reversible cycle can be decomposed into a bunch of Carnot “vortices” (more precisely, isothermal and adiabatic segments). Hence, establish the Clausius inequality:

\[\oint\frac{\overline{d}Q}{T}\leq 0\]

with equality if and only if the cycle is reversible (e.g. a Carnot cycle).

Solution:

The fact that \(\oint\frac{\overline{d}Q}{T}=0\) in the space of reversible cycles implies that there exists a conservative field \(S\) called the entropy that only depends on the initial and final equilibrium states.

Now suppose that first electron \(e^-\) naturally “burrows” its way down to the ground state \(\textbf n=\textbf k=\textbf 0\) in order to minimize its energy \(E=0\). Now put a second electron \(e^-\) into the box. In reality, the two electrons \(e^-\) would by their mutual Coulomb repulsion run away from each other on a hyperbolic orbit so to speak, raising all energy levels \(E_{\textbf n}\). However, for now we shall ignore all interactions. Then basically we have right now an ideal electron gas of two \(e^-\) that don’t talk to each other. However, actually there is one fundamental quantum mechanical “interaction” so to speak between the two \(e^-\) that cannot be ignored; this is the Pauli exclusion interaction arising from the identical fermionic nature of the spin \(s=1/2\) electrons \(e^-\). Here, as typical, we incorporate spin in an ad hoc manner as just another good quantum number whose associated operators \(\textbf S^2,S_3\) commute with the free space Hamiltonian \(H=T\) of the box. So if the first electron \(e^-\) is in state \(|\textbf n=\textbf 0\rangle\otimes|\uparrow\rangle\), then the second electron \(e^-\), if it also wants to minimize its energy, would have to occupy the state \(|\textbf n=\textbf 0\rangle\otimes|\downarrow\rangle\) (so the total state of the \(2\)-body electron system would be \(|\textbf n=\textbf 0\rangle\otimes|\textbf n=\textbf 0\rangle\otimes\frac{1}{\sqrt 2}(|\uparrow\rangle\otimes|\downarrow\rangle-|\downarrow\rangle\otimes|\uparrow\rangle)\)). The third electron \(e^-\) would then have to live in state \(|\textbf n=(1,0,0)\rangle\otimes|\uparrow\rangle\) for instance and in general we can house \(2\Omega(E_{(1,0,0)})=12\) electrons \(e^-\) in the next energy level, and so forth according to the non-zero values of the scatter plot above. In particular, if one puts in \(N\sim 10^{23}\) electrons for this ideal free electron gas, one would basically fill up a (discrete) ball of electrons in \(\textbf k\)-space (called the Fermi sea) of some radius \(k_F\approx\left(\frac{3N}{8\pi}\right)^{1/3}\), with each lattice point holding two electrons of opposite spin. The spherical boundary \(S^2\) of the Fermi sea of the ideal free electron gas would be called its Fermi surface, whose states thus have momentum \(\hbar k_F\) (called the Fermi momentum) and energy \(E_F=\frac{\hbar^2k_F^2}{2m}\) (called the Fermi energy).

Metals vs. Insulators

In practice, we’re not interested in an empty lattice \(\Lambda=\emptyset\), but rather a non-empty lattice \(\Lambda\neq\emptyset\) such as an atomic lattice in a solid! And what’s more, electrons \(e^-\) don’t just “get added” by some external agent, but rather emerge naturally as the valence electrons \(\partial e^-\) from the atoms located at the lattice sites \(\textbf x\in\Lambda\); thus, working inside a solid kills both birds with one stone.

At a high level, the act of superimposing a Bravais lattice \(\Lambda\) within what used to be an empty cube of sides \(L\) can be treated perturbatively exactly as one does in the nearly free electron model. Specifically, we’re still keeping the ideality/non-interacting assumption between the electrons (with the caveat of the Pauli interaction already mentioned), but now we go from being free\(\rightarrow\)nearly free (used in a technical sense). From that analysis, we know that the presence of the \(\Lambda\)-periodic potential butchers the previous free electron dispersion relation \(E(\textbf k)=\hbar^2|\textbf k|^2/2m\) into an energy band structure \(E_{\text{bands}}(\textbf k)\) where each Brillouin zone \(\Gamma_{\Lambda^*}^{(0)},\Gamma_{\Lambda^*}^{(1)}\), etc. (or bijectively, each energy band \(E_{\text{bands}}(\Gamma_{\Lambda^*}^{(0)}),E_{\text{bands}}(\Gamma_{\Lambda^*}^{(1)})\)) can accommodate precisely \(N\) momentum \(\textbf k\)-states (and thus \(2N\) electrons \(e^-\)) where \(N\) is the number of atoms in the solid. Suppose each atom contributes \(Z\) valence electrons (called its valency). Then the total number of free electrons roaming the solid will be \(ZN\), corresponding to the occupation of \(Z/2\) Brillouin zones or equivalently energy bands. In the ideal free electron case, we saw that the Fermi sea was a ball and its Fermi surface boundary \(\partial=S^2\) a sphere. Now, in the ideal nearly free electron case, we know that the energy is lowered at the boundary \(\partial\Gamma_{\Lambda^*}\) of the Brillouin zones (viewed from “within” otherwise how would the energy band gap form?) so this will distort the Fermi sea (and by extension the Fermi surface \(E_F\)-“equipotential”) of electrons towards it (but conserving in this case the area or in \(3\)D the volume of the Fermi sea since that’s the number of occupied \(\textbf P\)-eigenstates). For \(Z=1\) alkali metal solids or other metals (e.g. \(\text{Li},\text{Cu}\)), the act of “turning” on the perturbation due to the presence of the lattice \(\Lambda\) would look (for a \(2D\) material) something like (because \(Z=1\), the area of the initial Fermi sea/disk must be half the area of the square Brillouin zone):

It is clear in this \(2\)-dimensional case that, if the potential \(V\) induces a sufficiently large energy band gap (as it turns out it does for \(\text{Cu}\)), the Fermi surface can cross the Brillouin zone boundary \(\partial\Gamma_{\Lambda^*}\), though it must do so orthogonally in the reduced scheme to maintain its smoothness by virtue of the toroidal topology \(\Gamma_{\Lambda^*}\cong S^1\times S^1\) of the Brillouin zone.

Now, why do we care about the Fermi surface so much? Short answer: because materials with Fermi surfaces are metals. Qualitatively, the idea is that only the electrons with momentum \(\textbf k\) living on the Fermi surface of the system can actually do anything (having access to a bunch of unoccupied states slightly higher in energy to be able to respond to e.g. an \(\textbf E_{\text{ext}}\) to minimize their energy and form a current \(\textbf J=\sigma\textbf E_{\text{ext}}\)) just as only the valence electrons \(\partial e^-\) could delocalize (except that the former is a “meta-layer” of valency above the latter!). Any electrons \(e^-\) deep in the Fermi sea are pretty much trapped there by the Pauli exclusion principle since there’s no room for them to climb up into nearby energy levels above because they’re already occupied by other electrons (it would take a lot of energy for them to escape)!

For a sense of scale, most metals typically have Fermi temperatures on the order of \(T_F=E_F/k\sim 10^4\text{ K}\) (which is about twice as hot as the surface of the Sun \(\odot\)). This is why Fermi surfaces are also strictly defined at absolute zero \(T=0\), since most real materials in room temperature environments are nowhere near their Fermi temperature \(T_F\).

Another point is that, of course, the number of low-energy excitations available in a metal is proportional to the surface area \(\sim k_F^2\sim E_F\) of its Fermi surface (because each point on the Fermi surface corresponds to an excitable electron \(e^-\)).

Now consider \(Z=2\) atoms (e.g. alkaline earth atoms like \(\text{Be}\)). Now the initial Fermi sea/disk for the ideal free electron system has area equal to the square, but geometrically this implies that it must leak out the boundary of the Brillouin zone a little bit. If one now superimposes the perturbing lattice \(\Lambda\), there are \(2\) possibilities depending on the strength of the periodic lattice potential \(V_{\Lambda}\):

One might think the Fermi surface in the case of band insulators is just the boundary \(\partial\Gamma_{\Lambda^*}\) of the Brillouin zone but that’s not right because it’s not an equipotential with respect to any well-defined Fermi energy \(E_F\).

From here onwards \(Z=3,4,5,…\), the qualitative classification essentially repeats. Metals for instance may have several fully-occupied core bands, a fully-occupied valence band, and then right above that a partially-occupied conduction band. Note though that the Fermi surface need not lie solely in a single Brillouin zone, but can have sections distributed through several Brillouin zones. For example, if we consider the ideal, free electron gas with \(Z=3\) this time (so the area of the circle is three times that of the square \(1\)-st Brillouin zone that it now contains), then:

Although the Fermi surface in the \(3\)-rd Brillouin zone square looks disconnected, in fact it is connected (as it was in the extended scheme) because opposite sides of that square in the reduced scheme are topologically glued together (put another way, within the \(3\)-rd Brillouin zone reduced square, if one shifts the “origin” from where it is right now to the top right corner for instance (although all corners are equivalent), then the Fermi surface clearly becomes connected (see this either by tessellating the square or thinking of it as a torus \(S^1\times S^1\) and wrapping the square on itself):

This concludes a qualitative overview of band theory (classification of materials based on whether they are gapped or gapless). While the framework in general is fairly robust, there are some situations where its predictions fail. And as one might suspect, the origin of such deviations are due to one of the assumptions of band theory not being satisfied, notably the assumption of ideality. Examples of these deviations include semiconductors, Mott insulators (cf. band insulators), and topological insulators.

The purpose of this post is to calculate the energy band structure of the famous \(2\)-D material graphene. This of course is a monolayer of carbon \(\text C\) atoms arranged in a hexagonal “honeycomb” lattice. Sheets of graphene stacked on each other are called graphite.

Triangular Lattices, Brillouin Zone, Dirac Points

The first thing to notice is that this hexagonal lattice is non-Bravais because the \(\text C\) atoms are not in identical environments. As usual, this is resolved by viewing it as the convolution of a Bravais triangular lattice \(\Lambda_{\Delta}\) with a motif of \(2\)-carbon atoms dropped at each lattice point of \(\Lambda_{\Delta}\) (strictly speaking, this \(\Lambda_{\Delta}\) that I’m calling “triangular” is also confusingly often called hexagonal in crystallography). Therefore, we use the following primitive lattice vectors \(\textbf x_1,\textbf x_2\) which \(\text{span}_{\textbf Z}(\textbf x_1,\textbf x_2)=\Lambda_{\Delta}\):

Note that the lengths of the primitive lattice vectors are related to the inter-carbon atom spacing \(a\approx 1.4\) Å by \(|\textbf x_1|=|\textbf x_2|=\sqrt 3a\) and that the area of the spanning parallelogram is \(|\textbf x_1\times\textbf x_2|=\frac{3\sqrt{3}}{2}a^2\). Of course we also have a reciprocal Bravais lattice \(\Lambda^*_{\Delta}\) which is also triangular and spanned by reciprocal lattice vectors \(\textbf k_1,\textbf k_2\in\Lambda^*_{\Delta}\):

where now \(|\textbf k_1|=|\textbf k_2|=\frac{4\pi}{3a}\) and \(|\textbf k_1\times\textbf k_2|=\frac{8\pi^2}{3\sqrt{3}a^2}\) by the defining criterion \(\textbf x_{\mu}\cdot\textbf k_{\nu}=2\pi\delta_{\mu\nu}\). The first Brillouin zone is hexagonal and henceforth we reduce all other Brillouin zones to this:

where the side lengths of the Brillouin zone hexagon are \(\frac{4\pi}{3\sqrt{3}a}\) and thus the area of the Brillouin zone is \(\frac{8\sqrt 3\pi^2}{a^2}\). It turns out for graphene that the corners of the Brillouin zone hexagon are especially interesting points, called the Dirac points \(\textbf k_{\text{Dirac}}^{\pm}\) of graphene. Despite the fact that the Brillouin zone hexagon has \(6\) vertices, there are only really \(2\) physically distinct Dirac points as labelled because the other Dirac points are manifestly connected to them by a reciprocal lattice vector in \(\text{span}_{\textbf Z}(\textbf k_1,\textbf k_2)=\Lambda^*_{\Delta}\) and so are identified modulo the reduced zone scheme.

Physics of Graphene (Tight-Binding)

Each carbon \(\text C\) atom in graphene has valence \(Z=1\) from donating its one \(2p_3\) atomic orbital electron \(e^-\) into a collective \(\pi\) orbital delocalized over the entire graphene sheet. We’ll model these electrons as being tightly bound to carbon \(\text C\) atoms but free to tunnel/hop around the graphene lattice. In general, the tight-binding/hopping Hamiltonian is:

\[H=H(\Lambda_{\Delta},\mathcal M,t_0,t_1,t_2,…)=-\sum_{n=0}^{\infty}t_n\sum_{\textbf x\in\Lambda_{\Delta}}\sum_{\textbf x’\in\Lambda_{\Delta}*\mathcal M:|\textbf x’-\textbf x|_1=n}|\textbf x\rangle\langle\textbf x’|+|\textbf x’\rangle\langle\textbf x|\]

where \(\mathcal M\) is the \(2\)-carbon atom motif at each lattice point in \(\Lambda_{\Delta}\), \(t_n\in\textbf R\) are hopping energy amplitudes for \(n\)-th nearest neighbours, \(|\textbf x’\rangle\langle\textbf x|=(|\textbf x\rangle\langle\textbf x’|)^{\dagger}\) are mutual adjoints which physically permits bidirectional tunneling and mathematically ensures \(H^{\dagger}=H\) is Hermitian, and \(|\textbf x-\textbf x’|_1\) is a taxicab-like metric on the triangular Bravais lattice \(\Lambda_{\Delta}\) which makes it into a metric space, defined as the shortest number of hops between the two points \(\textbf x,\textbf x’\in\Lambda_{\Delta}\) (in graph theoretic terms, this is commonly called the graph geodesic distance between \(\textbf x\) and \(\textbf x’\)). In practice, it is common to restrict to only \(n=0\) (the on-site energy) and \(n=1\) (nearest neighbour) hopping, thereby ignoring all hopping across \(n\geq 2\) atoms. For graphene, the nearest-neighbour hopping energy amplitude turns out to be \(t_1\approx 2.8\text{ eV}\). With this simplification, the tight-binding Hamiltonian becomes (via a resolution of the identity):

\[H\approx -2t_01-t_1\sum_{\textbf x\in\Lambda_{\Delta}}\sum_{\textbf x’\in\Lambda_{\Delta}*\mathcal M:|\textbf x’-\textbf x|_1=1}|\textbf x\rangle\langle\textbf x’|+|\textbf x’\rangle\langle\textbf x|\]

Since the \(n=0\) term \(-2t_01\) is just proportional to the identity \(1\), it doesn’t affect any of the physics so we can ignore it henceforth. For graphene, there are of course \(3\) nearest neighbour carbon atoms \(\textbf x’\in\Lambda_{\Delta}*\mathcal M\) for each carbon atom at position \(\textbf x\in\Lambda_{\Delta}\):

\[H=-t_1\sum_{\textbf x\in\Lambda}|\textbf x\rangle\langle\textbf x+\frac{2}{3}\textbf x_1-\frac{1}{3}\textbf x_2|+|\textbf x\rangle\langle\textbf x-\frac{1}{3}\textbf x_1+\frac{2}{3}\textbf x_2|+|\textbf x\rangle\langle\textbf x-\frac{1}{3}\textbf x_1-\frac{1}{3}\textbf x_2|\]

As usual, we now declare that we wish to solve \(H|E\rangle=E|E\rangle\). For this we’ll simply make a Wannier-type ansatz of the \(H\)-eigenstate \(|E\rangle\) as a sum of plane waves modulated by

\[\]

The purpose of this post is to discuss historically one of the first decision problems for which quantum computing was shown to provide an exponential advantage over classical computing. One of the initially striking discrepancies between classical logic gates such as \(\text{AND}\) and quantum logic gates \(\Gamma\in U\left(\textbf C^{2^N}\right)\) is that the former need not be invertible or reversible (though they can be like the \(\text{NOT}\) gate) whereas the latter are always invertible being unitary \(\Gamma^{-1}=\Gamma^{\dagger}\). However, this is a bit of an illusion. Actually, given any classical logic gate \(\Gamma:\{0,1\}^n\to\{0,1\}^m\) mapping input \(n\)-bit strings to output \(m\)-bit strings (also called Boolean vector-valued functions), it is possible to construct a reversible classical logic gate \(\tilde{\Gamma}:\{0,1\}^{n+m}\to\{0,1\}^{n+m}\) defined as follows:

\[\tilde{\Gamma}(\xi_1+\xi_2):=\xi_1+\left(\xi_2\oplus\Gamma(\xi_1)\right)\]

for all input bit strings \(\xi_1\in\{0,1\}^n,\xi_2\in\{0,1\}^m\).

A qubit is any quantum system with a two-dimensional state space \(\mathcal H\cong\textbf C^2\). In particular because the state space is two-dimensional \(\dim\textbf C^2=2\), the Gram-Schmidt orthogonalization algorithm guarantees the existence of an orthonormal basis \(|0\rangle,|1\rangle\in\mathcal H\) of state vectors spanning \(\text{span}_{\textbf C}|0\rangle,|1\rangle=\mathcal H\) the entire state space \(\mathcal H\) of the qubit. For a larger system of \(N\in\textbf Z^+\) qubits, the corresponding state space \(\mathcal H_N\) is the \(N\)-fold tensor product \(\mathcal H_N\cong\left(\textbf C^2\right)^{\otimes N}\cong\textbf C^{2^N}\) of each qubit’s individual two-dimensional state space, with the dimension \(\dim\mathcal H_N\) of this composite Hilbert space \(\mathcal H_N\) growing exponentially \(\dim\mathcal H_N=2^N\) in the number \(N\) of qubits.

Bloch Sphere Representation of Physical Qubit States

One useful way to visualize physical qubit states \(|\psi\rangle\in P(\textbf C^2)\) (of an individual qubit) is via the Bloch sphere. The idea is to think of the qubit as being concretely realized by the spin angular momentum degree of freedom of a spin \(s=1/2\) isolated electron \(e^-\) whose position is fixed in an inertial frame in \(\textbf R^3\). Suppose one were to then make a measurement (e.g. with a suitable Stern-Gerlach filter) of the electron’s spin angular momentum \(\textbf S\) along some arbitrary direction \(\hat{\textbf n}\in S^2\) of one’s liking given in spherical coordinates by \(\hat{\textbf n}=\cos\phi\sin\theta\hat{\textbf i}+\sin\phi\sin\theta\hat{\textbf j}+\cos\theta\hat{\textbf k}\). Then for spin \(s=1/2\) quantum particles, there are precisely two possible (though not necessarily equiprobable) outcomes of such a measurement, namely \(m_s=\pm 1/2\) corresponding to an electron \(e^-\) either “spinning” aligned or anti-aligned with the direction \(\hat{\textbf n}\) of measurement. In particular, the electron’s spin state then collapses non-unitarily to the corresponding \(\hat{\textbf n}\cdot\textbf S\)-eigenstate which one can check are given by \(|m_s=1/2\rangle=\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle\) and \(|m_s=-1/2\rangle=-e^{-i\phi}\sin(\theta/2)|0\rangle+\cos(\theta/2)|1\rangle\) where \(|0\rangle,|1\rangle\) is the \(S_3\)-eigenbasis. In particular, the \(\hat{\textbf n}\)-aligned spin angular momentum eigenstate \(|m_s=1/2\rangle\) of \(\hat{\textbf n}\cdot\textbf S\) provides a general parameterization of an arbitrary electron spin state in \(\textbf C^2\) simply because the electron \(e^-\) has to be “spinning” along some direction \(\hat{\textbf n}\in S^2\) at all times (of course the same is true of the \(\hat{\textbf n}\)-antialigned spin angular momentum eigenstate \(|m_s=-1/2\rangle\), but it’s just more intuitive to work with the former). The map \(|m_s=1/2\rangle\mapsto\hat{\textbf n}\) is then the essence of the Bloch sphere \(S^2\) representation of physical qubit states.

As an aside, there is another nice way to visualize the Bloch mapping \(|m_s=1/2\rangle\mapsto\hat{\textbf n}\), which is a map from \(\textbf C^2\to S^2\). The idea is to decompose it as the composition of a Hopf fibration \(\textbf C^2\to\textbf C\cup\{\infty\}\) onto the extended complex plane \(\textbf C\cup\{\infty\}\), followed by an (inverse) stereographic projection onto an equatorial Riemann sphere \(\textbf C\cup\{\infty\}\to S^2\). Specifically, the Hopf fibration exploits the projective nature of \(P(\textbf C^2)\) by computing the ratio of probability amplitudes \(|\psi\rangle\mapsto\frac{\langle 0|\psi\rangle}{\langle 1|\psi\rangle}\) so in particular \(\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle\mapsto e^{-i\phi}\cot(\theta/2)\), and the rest is summarized in this picture:

This discussion has been implicitly focused on the pure quantum states of the qubit, although one could also allow more general mixed ensembles of pure states described by a density operator \(\rho\). In that case, it turns out points inside the Bloch sphere (called the Bloch ball \(|\hat{\textbf n}|<1\)) correspond precisely to such mixed ensembles.

where one can clearly see by setting \(\theta=\pi/2\) in the formula and \(\phi=0,\pi/2,\pi,3\pi/2\) that those cardinal points on the Bloch sphere correspond to the physical qubit states \(|+\rangle=\frac{1}{\sqrt 2}|0\rangle+\frac{1}{\sqrt 2}|1\rangle\), \(|-\rangle=\frac{1}{\sqrt 2}|0\rangle-\frac{1}{\sqrt 2}|1\rangle)\), \(|⟲\rangle=\frac{1}{\sqrt 2}|0\rangle+\frac{i}{\sqrt 2}|1\rangle\), and \(|⟳\rangle=\frac{1}{\sqrt 2}|0\rangle-\frac{i}{\sqrt 2}|1\rangle\) (here, the swirly-arrow notation ⟲,⟳ is actually more of a nod to thinking of the qubit states as photon \(\gamma\) polarization states, specifically as circularly polarized light).

“Qubit Strings” and Quantum Logic Gates

Given a classical bit string \(\xi=b_1b_2b_3…b_N\in\{0,1\}^N\) of length \(|\xi|=N\), the “qubit string” analog of it is given by the obvious isomorphism \(\{0,1\}^N\to\mathcal H_N\) into the state space \(\mathcal H_N\) of an \(N\)-qubit system defined by \(b_1b_2b_3…b_N\mapsto|b_1\rangle\otimes|b_2\rangle\otimes|b_3\rangle\otimes…\otimes|b_N\rangle\). Since there are of course \(2^N\) bit strings \(\xi\in\{0,1\}^N\) of length \(N\), this is indeed an isomorphism with its image defining the canonical unentangled basis \(\text{span}_{\textbf C}\left\{\otimes_{i=1}^N|b_i\rangle:b_i=0\text{ or }1\text{ for all }1\leq i\leq N\right\}=\mathcal H_N\) for \(\mathcal H_N\). In practice, it will often be convenient for “side quantum computations” to “pad” several additional qubits whose states are all initialized “spin-up” \(|0\rangle\) on the Bloch sphere, thus \(|b_1\rangle\otimes|b_2\rangle\otimes|b_3\rangle\otimes…\otimes|b_N\rangle\mapsto |b_1\rangle\otimes|b_2\rangle\otimes|b_3\rangle\otimes…\otimes|b_N\rangle\otimes|0\rangle\otimes|0\rangle\otimes…\otimes|0\rangle\).

Recall that in the circuit model \(\mathcal M_{\text{circuit}}\) of classical computation, the computational steps \(\Gamma_i:\{0,1\}^*\to\{0,1\}^*\) that might comprise a classical algorithm for computing some function \(f:\{0,1\}^*\to\{0,1\}^*\) were a universal set of logic gates like \(\text{AND},\text{OR},\text{NOT}\), etc. Earlier, we just saw that quantumly, we should replace \(N\)-bit strings by their analogous \(N\)-qubit states. What becomes the analog of the logic gates that form the backbone of classical computation in the circuit model? Well, now we just promote them to quantum logic gates acting on multi-qubit states in what should now be thought of as the circuit model of quantum computation. More precisely, an \(N\)-qubit quantum logic gate is any unitary operator in \(U\left(\textbf C^{2^N}\right)\) acting on \(N\)-qubit states.

For \(N=1\) qubit, continuing with the earlier example of a qubit realized via the spin angular momentum degree of freedom \(\textbf S\) of an electron \(e^-\), any unitary operator in \(U(\textbf C^2)\) must look like an intrinsic rotation \(e^{-i\Delta{\boldsymbol{\phi}}\cdot\textbf S/\hbar}\in U(\textbf C^2)\) of the electron \(e^-\) (which includes its “spin” axis \(\hat{\textbf n}\)) by an angular displacement \(\Delta{\boldsymbol{\phi}}\in\textbf R^3\). More precisely, for spin \(s=1/2\) quantum particles, we have \([\textbf S]_{|0\rangle,|1\rangle}^{|0\rangle,|1\rangle}=\frac{\hbar}{2}\boldsymbol{\sigma}\), so in the \(S_3\)-eigenbasis \(|0\rangle,|1\rangle\) we have the \(SU(2)\)-analog of Euler’s formula:

\[\left[e^{-i\Delta{\boldsymbol{\phi}}\cdot\textbf S/\hbar}\right]_{|0\rangle,|1\rangle}^{|0\rangle,|1\rangle}=e^{-i\Delta{\boldsymbol{\phi}}\cdot\boldsymbol{\sigma}/2}=\cos\left(\frac{\Delta\phi}{2}\right)1-i\sin\left(\frac{\Delta\phi}{2}\right)(\Delta\hat{\boldsymbol{\phi}}\cdot\boldsymbol{\sigma})\]

where \(\Delta{\boldsymbol{\phi}}=\Delta{\phi}\Delta\hat{\boldsymbol{\phi}}\). In particular, we have the following standard single-qubit quantum logic gates:

where as a special case of Euler’s \(SU(2)\)-formula we have for \(j=0,1,2,3\) that \(\sigma_j=ie^{-i\pi\sigma_j/2}\). However, the \(i\) in front is merely a global \(U(1)\) phase factor which is explicitly ignored in the Bloch \(S^2\) representation of physical qubit states. This is why it is accurate to think of both the Pauli and Hadamard quantum logic gates as encoding \(180^{\circ}=\pi\text{ rad}\) rotations of the qubit state on the Bloch sphere about different axes. Note of course that the Pauli gates are exactly identical to the usual \(\frak{su}\)\((2)\) Pauli matrices. In particular, the Pauli \(X\)-gate is the quantum analog of the classical \(\text{NOT}\) logic gate since when acting on the \(S_3\)-eigenbasis it swaps \(|0\rangle\leftrightarrow|1\rangle\) (of course \(X\) also acts on arbitrary \(\textbf C^2\) superpositions of \(|0\rangle\) and \(|1\rangle\) too on the Bloch \(S^2\) which is the novelty of quantum computing!). And the phase shift gate \(P_{\Delta\phi}\) is really a one-parameter subgroup of quantum logic gates parameterized by the azimuth \(\Delta\phi\in\textbf R\), for instance \(P_{\Delta\phi=\pi}=Z\) and one sometimes also sees the quantum logic gates \(S=P_{\Delta\phi=\pi/2}\) and \(T=P_{\Delta\phi=\pi/4}\). Note that a lot of this notation does clash with standard quantum mechanics notation on \(\textbf R^3\) where \(X,Y,Z\) are often used to denote the Cartesian components of the position observable \(\textbf X\) (although this is partially amended by the existence of the alternative notation \(X_1,X_2,X_3\)). However, an irreparable notational conflict lies in the Hadamard gate \(H=(X+Z)/\sqrt{2}\) where \(H\) is conventionally reserved for the Hamiltonian of a quantum system. For each of these quantum logic gates, there exists a corresponding circuit schematic symbol which (usually) consists simply of enclosing the letter of the gate into a rectangular box and adding an appropriate number of input and output qubit feeds, for example the Hadamard gate:

Just as the classical \(\text{AND}\) and \(\text{OR}\) logic gates accept input bit strings \(\xi\in\{0,1\}^2\) of length \(|\xi|=2\), so there are also two-qubit quantum logic gates in \(U(\textbf C^2\otimes\textbf C^2)\). Here we list a few standard ones:

• CX Gate: just as the Pauli \(X\)-gate can be thought of as a quantum NOT gate, the CX gate (called the controlled NOT gate) is a controlled version of the Pauli \(X\)-gate. More precisely, when acting on the unentangled \(2\)-qubit basis \(|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle\) (also called “the” computational basis for \(2\)-qubit systems), the input state \(|b_1\rangle\) of a control qubit is acted upon by the identity \(1\) (i.e. nothing happens to it) whereas the input state \(|b_2\rangle\) of a second target qubit may or may not be acted upon by the Pauli \(X\)-gate \(|b_2\rangle\mapsto X|b_2\rangle\) depending on the control qubit’s input state \(|b_1\rangle\). Such a “truth table” can be expressed as \(CX|b_1\rangle\otimes|b_2\rangle:=|b_1\rangle\otimes X^{b_1}|b_2\rangle=|b_1\rangle\otimes|b_1\text{ XOR } b_2\rangle\) for \(b_1,b_2\in\{0,1\}\), thus showing that the controlled NOT gate \(CX\) can also be viewed as a sort of quantum analog of the classical exclusive or (XOR) logic gate. Note that the action of the controlled NOT gate \(CX\) on arbitrary states in \(\textbf C^{2}\otimes\textbf C^2\) follows from the above truth table by linearity. The controlled NOT gate \(CX\) thus also has matrix representation:

\[[CX]_{|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle}^{|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle}=\text{diag}(1_{2\times 2},[X]_{|0\rangle,|1\rangle}^{|0\rangle,|1\rangle})=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{pmatrix}\]

Viewed as a rotation on some kind of entangled Bloch sphere, one also has \(CX=e^{\pm i\frac{\pi}{4}(1-Z_1)\otimes(1-X_2)}\); this can be checked by a direct computation but I still don’t feel I have a strong intuition for why this is right, in particular why it should involve the Pauli \(Z\)-gate of the control qubit?

• CY Gate: The controlled Pauli \(Y\)-gate is conceptually identical to the definition of the controlled NOT gate (also called the controlled Pauli \(X\)-gate) just with \(X\mapsto Y\) everywhere in the discussion. The upshot is that \(CY|b_1\rangle\otimes|b_2\rangle=|b_1\rangle\otimes Y^{b_1}|b_2\rangle\) or equivalently:

\[[CY]_{|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle}^{|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle}=\text{diag}(1_{2\times 2},[Y]_{|0\rangle,|1\rangle}^{|0\rangle,|1\rangle})=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-i\\0&0&i&0\end{pmatrix}\]

The controlled Pauli \(Y\)-gate also has a Bloch sphere representation.

• CZ Gate: The controlled Pauli \(Z\)-gate is again the same idea (i.e. the idea of having a control qubit to control whether or not a target qubit is acted on by the Pauli \(Z\)-gate), i.e. \(CZ|b_1\rangle\otimes|b_2\rangle=|b_1\rangle\otimes Z^{b_1}|b_2\rangle\) or:

\[[CZ]_{|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle}^{|0\rangle\otimes|0\rangle, |0\rangle\otimes|1\rangle, |1\rangle\otimes|0\rangle, |1\rangle\otimes|1\rangle}=\text{diag}(1_{2\times 2},[Z]_{|0\rangle,|1\rangle}^{|0\rangle,|1\rangle})=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&-1\end{pmatrix}\]

The controlled Pauli \(Z\)-gate also has a Bloch sphere representation.

Measurement

Recall that in quantum mechanics, given an observable \(\textbf X\) (e.g. position) with spectrum \(\textbf x_1,\textbf x_2,…\) and a quantum system in some arbitrary state \(|\psi\rangle\), the Born rule asserts that each \(\textbf x_j\) has probability \(|\langle\textbf x_j|\psi\rangle|^2\) of being the outcome of an \(\textbf X\)-measurement. Specific to the Copenhagen interpretation of quantum measurement is that the state \(|\psi\rangle\) also collapses \(|\psi\rangle\mapsto|\textbf x_j\rangle\) to the \(\textbf X\)-eigenstate \(|\textbf x_j\rangle\) associated to the measured value \(\textbf x_j\). In other words, the measurement of the state \(|\psi\rangle\) randomly projects \(|\psi\rangle\) onto one of the eigenstates of \(\textbf X=\textbf x_1|\textbf x_1\rangle\langle\textbf x_1|+\textbf x_2|\textbf x_2\rangle\langle \textbf x_2|+…\) (this is \(|\psi\rangle\mapsto|\textbf x_j\rangle\langle\textbf x_j|\psi\rangle\cong|\textbf x_j\rangle\)) . However, regardless of the outcome, this measurement process is non-unitary \(\notin U(\mathcal H)\) because a projection \(P^2=P\Rightarrow\det P=0\), being irreversible/non-invertible, must be non-unitary (with the \(\det P=1\) exception that if a quantum system already happens to be in some \(\textbf X\)-eigenstate \(|\textbf x_j\rangle\), then in that case an \(\textbf X\)-measurement simply collapses the state via the identity \(|\textbf x_j\rangle\mapsto|\textbf x_j\rangle\) which is trivially unitary). Another way to convince yourself why measurement is non-unitary is that the inner products \(\langle\psi_1|\psi_2\rangle\mapsto\langle \textbf x_1|\textbf x_2\rangle=\delta_{\textbf x_1,\textbf x_2}\) are not necessarily \(1\) after the measurement (the particular pair of states only has probability \(\text{Tr}()=\sum_{\textbf x\in\Lambda_{\textbf X}}|\langle\textbf x|\psi_1\rangle|^2|\langle\textbf x|\psi_2\rangle|^2\) of being unitary in this sense, assuming they’re unentangled).

Despite being non-unitary, whereas we defined quantum logic gates to be unitary, nothing says we can’t still exploit measurements to our advantage in quantum computation so that in a way they sort of are like just another kind of quantum logic gate (albeit non-unitary and non-deterministic). More precisely, since the measurement will essentially give some concrete bit string, it is common to perform classical computations with the results of measurements; however, note that this doesn’t actually add any novel computational power to quantum computing.

In analogy to the notion of a universal set of logic gates in the circuit model of classical computation, one has the notion of an approximately universal set of quantum logic gates; we want to only consider some finite collection of quantum logic gates as somehow “generating” via compositions all possible quantum logic gates in \(\bigcup_{N\in\textbf N}U\left(\textbf C^{2^N}\right)\), but the group \(\bigcup_{N\in\textbf N}U\left(\textbf C^{2^N}\right)\) has uncountably infinite order whereas any finite collection of quantum logic gates in \(\bigcup_{N\in\textbf N}U\left(\textbf C^{2^N}\right)\) clearly can only generate a countably infinite subgroup (this is similar to how the cyclic subgroup \(C_{\infty}=\{R^n:n\in\textbf N\}\) of \(SO(2)\) generated by some rotation matrix \(R\in SO(2)\) does not actually generate \(SO(2)\) itself, i.e. \(C_{\infty}\neq SO(2)\) again on order grounds \(|C_{\infty}|=\aleph_0<|SO(2)|\) though \(C_{\infty}\) will be dense in \(SO(2)\) provided the generator \(R=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\) rotates \(\textbf R^2\) through an irrational angle \(\theta\in\textbf R-\textbf Q\)).

Definition (Approximate Universality): A collection of quantum logic gates \(\mathcal G\subseteq\bigcup_{N\in\textbf N}U\left(\textbf C^{2^N}\right)\) is said to be approximately universal iff for arbitrarily small \(\varepsilon>0\) and any quantum logic gate \(\Gamma\in\bigcup_{N\in\textbf N}U\left(\textbf C^{2^N}\right)\), there exists some quantum logic circuit \(\Gamma_1\circ\Gamma_2\circ…\) with each \(\Gamma_i\in\mathcal G\) such that \(|\Gamma-\Gamma_1\circ\Gamma_2\circ…|<\varepsilon\) in the operator norm, or equivalently \(\sup_{\langle\psi|\psi\rangle=1}|\Gamma|\psi\rangle-\Gamma_1\circ\Gamma_2\circ…|\psi\rangle|<\varepsilon\).

For instance, it can be checked that the collections \(\{\text{CX}\}\cup U(\textbf C^2)\) and \(\{CX,H,T\}\) are approximately universal sets of quantum logic gates. In fact, the former, being uncountably infinite, is actually an exactly universal set of quantum logic gates which means what it sounds like (i.e. \(\varepsilon=0\)).

Mechanical Model of Measurement

I wonder if one can build a mechanical model to illustrate this, e.g. a flashlight mounted on a spinner for an \(\textbf S\)-measurement of a qubit.

Quantum Complexity Classes

Recall that \(\textbf{BPP}\), called the bounded error probabilistic polynomial time complexity class, is the classical complexity class of all decision problems (or equivalently their binary languages \(\mathcal L\subseteq\{0,1\}^*\)) for which there exists a randomized polynomial-time algorithm with “better-than-random” probability of correctly computing the indicator function \(\in_{\mathcal L}:\{0,1\}^*\to\{0,1\}\) on each input bit string \(\xi\in\{0,1\}^*\). The quantum complexity class analog of the classical \(\textbf{BPP}\) complexity class is called \(\textbf{BQP}\), standing for bounded error quantum polynomial time, representing the set of all decision problems for which there exists a polynomial-time quantum circuit/algorithm using some particular approximately universal collection of quantum logic gates that correctly computes the answer to the decision problem at least \(2/3\) of the time (again, the \(2/3\) is sort of arbitrary).

One can check that \(\textbf{BQP}\) is independent of which particular collection of approximately universal quantum logic gates one uses. Just as classically, we have the quantum analog of Cobham’s thesis that \(\textbf{BQP}\) is the class of all “feasible quantum decision problems”. Indeed, there is a simple connection known between \(\textbf{BPP}\) and \(\textbf{BQP}\), namely that \(\textbf{BPP}\subseteq\textbf{BQP}\). The question of whether or not quantum computing is strictly more powerful than classical computing can be phrased roughly as the question: “is \(\textbf{BPP}=\textbf{BQP}\)”? If true, this would mean the answer is “no”.

The purpose of this post is to quickly review some fundamentals of classical computation in order to better appreciate the distinctions between classical computing and quantum computing. Note that the word computation itself, whether classical or quantum, basically just means evaluating functions \(f:\alpha^*\to\alpha^*\) defined on the Kleene closure \(\alpha^*\) of an arbitrary alphabet/set \(\alpha\) (recall that sequences of symbols from the alphabet \(\alpha\) are called strings over \(\alpha\) with \(\alpha^*=\bigcup_{n\in\textbf N}\alpha^n\) the set of all strings over \(\alpha\)). However, for any at most countable alphabet \(|\alpha|\in\textbf N\cup\{\aleph_0\}\), the Kleene closure \(\alpha^*\) will be countably infinite \(|\alpha^*|=\aleph_0\), so it should therefore be possible to exhibit a injection \(\alpha^*\to\alpha_{\text{binary}}^*\) from strings in \(\alpha^*\) to bit strings in \(\alpha_{\text{binary}}^*\) where \(\alpha_{\text{binary}}:=\{0,1\}\) is the binary alphabet. In turn, this simply means finding a (binary) encoding \(\mathcal E:\alpha\to\alpha_{\text{binary}}^*\) at the level of the individual symbols in the alphabet \(\alpha\) into bit strings in \(\alpha_{\text{binary}}^*\) and then concatenating together bit strings for individual symbols in \(\alpha\) to obtain bit strings for strings in \(\alpha^*\). In image processing for instance, such an encoding \(\mathcal E\) would be thought of as lossless compression of the data/symbols in the alphabet \(\alpha\), and there are many possible encoding schemes depending on the situation, for instance a fixed-length encoding \(\mathcal E_{\text{FL}}\), or variable-length encodings such as run-length encoding \(\mathcal E_{\text{RL}}\) or Huffman encoding \(\mathcal E_{\text{Huffman}}\). If one uses a fixed-length encoding \(\mathcal E_{\text{FL}}\), then the overhead cost associated with \(\mathcal E_{\text{FL}}\) is linear \(\mathcal E_{\text{FL}}(a_1+a_2)=\mathcal E_{\text{FL}}(a_1)+\mathcal E_{\text{FL}}(a_2)\) for all symbols \(a_1,a_2\in\alpha\) (where here we adapt the Pythonic meaning of \(+\) as a string concatenation; note that the encoding \(\mathcal E_{\text{FL}}\) is technically only defined on the alphabet \(\alpha\), so the statement above is more of a definition of how to extend the domain of \(\mathcal E_{\text{FL}}\) from \(\alpha\) to \(\alpha^*\)). This discussion is just to say that henceforth we can assume without loss of generality that the alphabet \(\alpha=\alpha_{\text{binary}}\) is just the binary alphabet, and so all computations will be concerned with evaluating functions \(f:\{0,1\}^*\to\{0,1\}^*\) which map input bit strings to output bit strings.

Defining a binary language \(\mathcal L\) to be any collection of bit strings \(\mathcal L\subseteq\{0,1\}^*\), the decision problem for \(\mathcal L\) is to compute the indicator function \(\in_{\mathcal L}:\{0,1\}^*\to\{0,1\}^*\) of \(\mathcal L\) defined for all input bit strings \(\xi\in\{0,1\}^*\) by the Iverson bracket \(\in_{\mathcal L}(\xi):=[\xi\in\mathcal L]\). Note however that although in general we allow the codomain \(\{0,1\}^*\) of the function \(\in_{\mathcal L}\) being computed to consist of output bit strings of arbitrary length, in fact for binary language decision problems the range \(\in_{\mathcal L}(\{0,1\}^*)\) as manifest in the Iverson bracket is restricted to the \(2\) possible one-bit strings \(\in_{\mathcal L}(\{0,1\}^*)=\{0,1\}\subseteq\{0,1\}^*\). A very important example of a decision problem is primality testing where the binary language \(\mathcal L:=\{10,11,101,111,1011,…\}\) consists of all the prime numbers. The decision problem of primality testing is a decidable decision problem by virtue of e.g. the AKS primality test. On the contrary, there unfortunately do also exist undecidable decision problems, examples being the halting problem, Hilbert’s tenth problem, etc. The notion of decidability is specific to decision problems and generalizes to non-decision problems via the notion of computability.

Circuit Model of Classical Computation

Recall that a computation is an evaluation of a function \(f:\{0,1\}^*\to\{0,1\}^*\). A computational model \(\mathcal M\) is any collection \(\mathcal M=\{\Gamma_i:\{0,1\}^*\to\{0,1\}^*|\Gamma_i(\xi)\in O_{|\xi|\to\infty}(1)\}\) of computational steps \(\Gamma_i\in\mathcal M\) considered legal within the framework of \(\mathcal M\), and any well-defined sequence \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…=f\) of such computational steps that compose to the function \(f\) of interest is called an algorithm for computing \(f\) in \(\mathcal M\). Historically, many computational models \(\mathcal M\) (e.g. Turing machines, general recursive functions, lambda calculus, Post machines, register machines) have been proposed in which the nature of the allowed computational steps \(\Gamma_i\in\mathcal M\) seem a priori to give rise to different classes of computable functions \(f\) (i.e. functions \(f\) for which there even exists \(\exists\) some algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…=f\) for computing \(f\)). However, the Church-Turing thesis is an informal conjecture/axiom in computability theory which roughly asserts that any “reasonable” computational model \(\mathcal M\) one can dream up will in fact have neither greater nor less computing power compared with any other “reasonable” computational model \(\mathcal M’\cong\mathcal M\). Thus, for the purposes of a smoother transition from classical computing to quantum computing, we will henceforth take the circuit model \(\mathcal M_{\text{circuit}}\) to be our computational model. In the circuit model of classical computation \(\mathcal M_{\text{circuit}}\), the allowed computational steps are the \(3\) logic gates \(\mathcal M_{\text{circuit}}=\{\text{AND},\text{OR},\text{NOT}\}\) which form a universal set for arbitrary logic circuits/Boolean functions (though the \(\text{OR}\) gate is actually redundant! see here). For instance, binary adder/multiplier logic circuits are not considered to be computational steps in the circuit model \(\mathcal M_{\text{circuit}}\) because say for two input bit strings \(\xi_1,\xi_2\) of the same length \(n=|\xi_1|=|\xi_2|\) they are not \(O_{n\to\infty}(1)\) as required.

Randomized Classical Computational Models

In anticipation of the randomness inherent in outputs of quantum computations, it will be helpful to quickly review what it means for a classical computational model (such as the circuit model of classical computation) to be randomized. This means that when computing a function \(f:\{0,1\}^*\to\{0,1\}^*\), any input bit string \(\xi\in\{0,1\}^*\) is also concatenated \(\xi\mapsto \xi+\xi_{\text{random}}\) with a uniformly random input bit string from a hypothetical random number generator such as \(\xi_{\text{random}}=011011010001001\in\{0,1\}^*\) (so for each run of the computation, even for the same input bit string \(\xi\), the input bit string \(\xi_{\text{random}}\) will probably be different from the previous run and thus the output bit string of the computation will be a sample from a probability distribution) but otherwise one now just chugs the total input bit string \(\xi+\xi_{\text{random}}\) through some sequence of computational steps \(\Gamma_1,\Gamma_2,…\) within that computational model. For instance, in the circuit model, if one desires a particular logic gate \(\Gamma\) in a computation to act with \(50\%\) probability as an \(\text{AND}\) gate and with \(50\%\) probability as an \(\text{OR}\) gate, then one can define \(\Gamma:\{0,1\}^3\to\{0,1\}\) to take an input bit string \(b_1b_2b_3\) of length \(3\) (i.e. \(3\) input bits) but where say the third input bit \(b_3\) is one of the random input bits in the string \(\xi_{\text{random}}\) and determines the nature of the logic gate \(\Gamma\) according to:

\[\Gamma(b_1,b_2,b_3):=\begin{cases}
\text{AND}(b_1,b_2) & \text{if } b_3 = 0 \\
\text{OR}(b_1,b_2) & \text{if } b_3 = 1
\end{cases}\]

Among randomized computational models, there are \(2\) subclasses: Las Vegas algorithms such as the quicksort algorithm which emphasize correctness of the computational output bit string and Monte Carlo “algorithms” such as Miller-Rabin primality testing which emphasize high probability of correctness of the computational output bit string.

Finally, a note about determinism in computational models. Classical computational models are always deterministic because \(f\) is a function. I would argue that even randomized classical computational models are deterministic because, if for two runs the exact same random bit string \(\xi_{\text{random}}\) happens to be inputted, then the output bit string \(f(\xi+\xi_{\text{random}})\in\{0,1\}^*\) must deterministically be the same. So what would be a non-deterministic computational model? Basically, the God-like magical power of being able to just instantaneously “guess” the correct output bit string \(f(\xi)\in\{0,1\}^*\) for any input bit string \(\xi\in\{0,1\}^*\). Such a non-deterministic computer cannot exist, but is an important theoretical construct for defining the nondeterministic polynomial time complexity class \(NP\) which will be elaborated in the following section.

Classical Complexity Classes

If, in some computational model \(\mathcal M\), someone has found an algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…=f\) for computing \(f\) (hopefully correctly or at least with high probability of being correct such as in Monte Carlo “algorithms”), then one can always ask about how efficient the algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\) is at computing \(f\). This is, in computer science lingo, a question of the algorithm’s complexity. Associated to any algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\) are \(2\) distinct kinds of complexity: time complexity (i.e. the number of computational steps \(\Gamma_i\) in the algorithm sequence \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\)) and space complexity (i.e. “amount” of memory/”RAM” used during execution of the algorithm sequence \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\)). In general, both the time complexity and space complexity of an algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\) are to be viewed as functions of which particular input bit string \(\xi\in\{0,1\}^*\) is fed into the algorithm \(\xi\mapsto\Gamma_1\circ\Gamma_2\circ\Gamma_3…(\xi)\) where in general, input bit strings \(\xi\in\{0,1\}^*\) of longer length \(|\xi|\) may be expected to require greater time complexity \(T(\xi)\) and space complexity \(Sp(\xi)\). More precisely, one is often interested in the behavior of the worst-case time complexity \(T^*(n):=\sup_{\xi\in\{0,1\}^n}T(\xi)\) of the algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\) in the asymptotic \(n\to\infty\) limit of longer and longer input bit strings \(\xi\in\{0,1\}^*\) and likewise the behavior of the worst-case space complexity \(Sp^*(n):=\sup_{\xi\in\{0,1\}^n}Sp(\xi)\) in the asymptotic \(n\to\infty\) limit. Sometimes though it’s also interesting to analyze the algorithm’s average time complexity \(\bar T(n):=\frac{1}{2^n}\sum_{\xi\in\{0,1\}^n}T(\xi)\) or the average space complexity \(\bar{Sp}(n):=\frac{1}{2^n}\sum_{\xi\in\{0,1\}^n}Sp(\xi)\), both asymptotically \(n\to\infty\).

Example: For a list of length \(n\in\textbf N\), the linear search algorithm has (both worst-case and average) time complexity that scales asymptotically as \(T^*_{\text{linear search}}(n),\bar T_{\text{linear search}}(n)\in O_{n\to\infty}(n)\) but (both worst-case and average) space complexity that scales asymptotically as \(Sp^*_{\text{linear search}}(n),\bar{Sp}_{\text{linear search}}(n)\in O_{n\to\infty}(1)\). By contrast, the merge sort algorithm has (both worst-case and average) time complexity that scales asymptotically as \(T^*_{\text{merge sort}}(n),\bar T_{\text{merge sort}}(n)\in O_{n\to\infty}(n\log n)\) and (both worst-case and average) space complexity that scales asymptotically as \(Sp^*_{\text{merge sort}}(n),\bar{Sp}_{\text{merge sort}}(n)\in O_{n\to\infty}(n)\). This is obvious, but note that searching a list vs. sorting a list are different computations so one cannot just directly compare these algorithm’s asymptotic spacetime complexities to declare linear search is “more efficient” than merge sort; however binary search is indeed an asymptotically more efficient search algorithm than linear search as measured by their spacetime complexities in the worst-case or averaged senses.

Recall that a decision problem concerns the computation of the indicator function \(\in_{\mathcal L}:\{0,1\}^*\to\{0,1\}\) of a binary language \(\mathcal L\subseteq\{0,1\}^*\) of bit strings. To this effect, we define:

Definition (\(\textbf P\)): The polynomial time complexity class \(\textbf P\) is defined to be the set of all binary languages \(\mathcal L\), or equivalently decision problems, for which there exists an algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…=\in_{\mathcal L}\) that computes the indicator function \(\in_{\mathcal L}:\{0,1\}^*\to\{0,1\}\) with worst-case time complexity \(T^*_{\Gamma_1\circ\Gamma_2\circ\Gamma_3…}(n)\in\bigcup_{p\in\textbf N}O_{n\to\infty}(n^p)\).

Notice in the definition that we haven’t specified a computational model \(\mathcal M\) from which we derive the computational steps \(\Gamma_i\in\mathcal M\). Indeed, it turns out that the polynomial time complexity class \(\textbf P\) (as with all other complexity classes) is robust with respect to whichever computational model \(\mathcal M\) one chooses to use in the sense that a given binary language/decision problem is in \(\textbf P\) in one computational model \(\mathcal M\) if and only if it’s in \(\textbf P\) in any other Church-Turing equivalent computational model \(\mathcal M’\). Also, note that the polynomial time complexity class \(\textbf P\) doesn’t a priori impose any constraints on the (worst-case or average) space complexity \(Sp_{\Gamma_1\circ\Gamma_2\circ\Gamma_3…}(n)\) of the algorithm. Nonetheless, one can show that \(\textbf P\subseteq \textbf{PSPACE}\) where \(\textbf{PSPACE}\) is the polynomial space complexity class defined verbatim to the polynomial time complexity class \(\textbf P\) except replacing the word “time” with “space” and thus \(T\mapsto Sp\). Whether or not the reverse inclusion \(\textbf{PSPACE}\subseteq \textbf{P}\) is true is still an open question!

Examples: Primality testing is a decision problem which recently (thanks to the AKS primality test) has been shown to be in the polynomial time complexity class \(\textbf P\) (and therefore also in \(\textbf{PSPACE}\)). Note that a naive primality test in which we take a given \(N\in\textbf Z^+\) and divide it by each positive integer \(\leq\sqrt{N}\) does not constitute a polynomial-time algorithm for primality testing. This is because \(\sqrt{N}=2^{\log_2(N)/2}\), but \(n:=\log_2(N)\) is the actual length of the input bit string for \(N\) so therefore we see that this naive primality test algorithm \(T^*_{\text{naive primality test}}(n)\in O_{n\to\infty}(2^n)\) is actually exponential-time. Although it may seem that list searching/ sorting problems are also members of the polynomial time complexity class \(\textbf P\) (thanks to e.g. linear search, binary search, merge sort, etc.), this is strictly speaking only the case if one reframes the original search/sorting problem into a decision problem like “does this list contain the bit string \(1011\)” so that the binary language \(\mathcal L=\{1011\}\) is singleton. Otherwise, one would more accurately consider list searching/sorting problems to be in the complexity class \(\textbf{FP}\) of “functions”.

Owing to the existence of randomized computation leading to randomized algorithms (in particular Monte Carlo algorithms), we also have:

Definition (\(\textbf{BPP}\)): The bounded error probabilistic polynomial time complexity class \(\textbf{BPP}\) is defined to be the set of all binary languages \(\mathcal L\), or equivalently decision problems, for which there exists a randomized algorithm \(\Gamma_1\circ\Gamma_2\circ\Gamma_3…\approx \in_{\mathcal L}\) that attempts to compute the indicator function \(\in_{\mathcal L}:\{0,1\}^*\to\{0,1\}\) with worst-case time complexity \(T^*_{\Gamma_1\circ\Gamma_2\circ\Gamma_3…}(n)\in\bigcup_{q\in\textbf N}O_{n\to\infty}(n^q)\) and correctness probability \(p(\xi)\in(1/2,1]\) for all input bit strings \(\xi\in\{0,1\}^*\) (commonly just declared to be \(\geq 2/3\), though this is arbitrary because one can consider a new algorithm defined by running arbitrarily many Bernoulli trials of the original algorithm, taking the majority decision, and applying the Chernoff bound; thus it suffices to find a Monte Carlo algorithm with \(p(\xi)\in(1/2,1)\) or a Las Vegas algorithm with \(p(\xi)=1\)).

According to Cobham’s thesis, the bounded error probabilistic polynomial time complexity class \(\textbf{BPP}\) (with \(\textbf P\subseteq\textbf{BPP}\) obviously contained therein) of decision problems are considered “classically feasible decision problems” and so the goal of computer scientists is to prove that various \(\mathcal L\)-decision problems are in \(\textbf{BPP}\) by devising clever (possibly randomized) polynomial-time algorithms to compute their corresponding indicator functions (either always correctly or more often than not correctly!). Decision problems which are not in \(\textbf{BPP}\) (or more precisely, haven’t been shown to be in \(\textbf{BPP}\) because no one has discovered a polynomial-time algorithm to compute their indicator functions yet) are therefore the hardest decision problems to solve. Among these are:

Definition (\(\textbf{NP}\)): The nondeterministic polynomial time complexity class \(\textbf{NP}\) is defined to be the set of all binary languages \(\mathcal L\), or equivalently decision problems, for which there exists a nondeterministic algorithm \(\Gamma_{\text{God}}=\in_{\mathcal L}\) that computes the indicator function \(\in_{\mathcal L}:\{0,1\}^*\to\{0,1\}\) with worst-case time complexity \(T^*_{\Gamma_{\text{God}}}(n)\in\bigcup_{p\in\textbf N}O_{n\to\infty}(n^p)\).

In other words, an \(\mathcal L\)-decision problem is in \(\mathcal L\in\textbf{NP}\) iff its corresponding \(\mathcal L\)-verification decision problem concerning the computation of the function \([\xi\in]\) is in \(\textbf P\). The following diagram summarizes the current state of knowledge as of \(2024\), October \(3\)rd (note that the relationship between \(\textbf{BPP}\) and \(\textbf{NP}\) is still not known!):

Problem #\(1\): What does the phrase “\(N\) coupled harmonic oscillators” mean?

Solution #\(1\): Basically, just think of \(N\) masses \(m_1,m_2,…,m_N\) with some arbitrarily complicated network of springs (each of which could have different spring constants) connecting various pairs of masses together:

Problem #\(2\): Rephrase Solution #\(1\) in more mathematical terms.

Solution #\(2\): The idea is that there are \(N\) degrees of freedom \(x_1,x_2,…,x_N\) and the second time derivative of each one is given by some linear combination of the \(N\) degrees of freedom; packaging into a vector \(\textbf x:=(x_1,x_2,…,x_N)^T\in\textbf R^N\), this means that there exists an \(N\times N\) matrix \(\omega^2\) (independent of \(t\)) such that:

\[\ddot{\textbf x}=-\omega^2\textbf x\]

Problem #\(3\): In general, how does one solve the second-order ODE in Solution #\(2\)?

Solution #\(3\): One way is to first rewrite it as a first-order ODE:

\[\frac{d}{dt}\begin{pmatrix}\textbf x\\\dot{\textbf x}\end{pmatrix}=\begin{pmatrix}0&1\\-\omega_0^2&0\end{pmatrix}\begin{pmatrix}\textbf x\\\dot{\textbf x}\end{pmatrix}\]

from which the solution is immediate:

\[\begin{pmatrix}\textbf x(t)\\\dot{\textbf x}(t)\end{pmatrix}=e^{t\begin{pmatrix}0&1\\-\omega^2&0\end{pmatrix}}\begin{pmatrix}\textbf x(0)\\\dot{\textbf x}(0)\end{pmatrix}\]

and using \(\begin{pmatrix}0&1\\-\omega^2&0\end{pmatrix}^2=-\begin{pmatrix}\omega^2&0\\0&\omega^2\end{pmatrix}\), one has the self-consistent solutions:

\[\textbf x(t)=\cos(\omega t)\textbf x(0)+\omega^{-1}\sin(\omega t)\dot{\textbf x}(0)\]

\[\dot{\textbf x}(t)=-\omega\sin(\omega t)\textbf x(0)+\cos(\omega t)\dot{\textbf x}(0)\]

Problem #\(4\): In practice how would one evaluate the quantities \(\cos(\omega t)\) and \(\omega^{-1}\sin(\omega t)\) appearing in Solution #\(3\)?

Solution #\(4\): One has:

\[\cos(\omega t)=1-\frac{\omega^2 t^2}{2}+\frac{\omega^4t^4}{24}-…\]

\[\omega^{-1}\sin(\omega t)=1-\frac{\omega^2 t^3}{6}+\frac{\omega^4t^5}{120}-…\]

The point is that one needs an efficient way to evaluate powers \(\omega^{2n}\) of the matrix \(\omega^2\) appearing in the equation of motion \(\ddot{\textbf x}=-\omega^2\textbf x\); the standard way to do this is to diagonalize \(\omega^2\).

Problem #\(5\): Interpret the diagonalization of \(\omega^2\) in Solution #\(4\) as a physicist.

Solution #\(5\): Each eigenspace of \(\omega^2\) is called a normal mode of the corresponding system of coupled harmonic oscillators, i.e. a normal mode should be thought of as a pair \((\omega_0,\ker(\omega^2-\omega_0^21))\); since \(\omega^2\) is an \(N\times N\) matrix, there will in general be \(N\) normal modes (assuming \(\omega^2\) is diagonalizable which in practice will always be the case). Viewed in this way, the general dynamics of \(N\) coupled harmonic oscillators consists of a superposition of the \(N\) normal modes of the system, with the coefficients in this superposition determined by the initial conditions \(\textbf x(0),\dot{\textbf x}(0)\).

Physically, because a normal mode is defined by a single eigenfrequency \(\omega_0\), it means that if one were to release the system with exactly the right initial conditions to excite only that one specific normal mode, then all \(N\) coupled harmonic oscillators would oscillate at the same frequency \(\omega_0\) with some relative amplitudes and relative phases between each of their oscillations as encoded in the corresponding eigenspace \(\ker(\omega^2-\omega_0^21)\).

Problem #\(6\): Determine the normal modes of the following systems of coupled harmonic oscillators:

Solution #\(6\): For the \(2\) masses connected to each other and to immoveable walls (equivalent to having infinite masses at those locations):

Needless to say for generic \(m_1,m_2,k_1,k_2,k_3\) the normal modes are not so straightforward either. There are of course also many variations of this setup, for instance \(k_3=0\) (i.e. just no wall there) or having \(m_1=m_2\) and \(k_1=k_2=k_3\), etc.

For the double pendulum, it is of course the standard example of a nonlinear dynamical system, but it has a stable fixed point at \(\theta_1=\theta_2=0\) so one can linearize the system about it to obtain \(2\) coupled harmonic oscillators:

Problem #\(7\): Consider extending the first system from Problem #\(6\) from \(N=2\) to \(N\gg 1\) identical masses \(m\) connected by identical springs \(k_s\); in condensed matter physics this is known as the \(1\)D monatomic chain, a classical toy model of a solid. Write down the equation of motion for the displacement \(x_n(t)\) of the \(n\)th atom about its equilibrium where \(2\leq n\leq N-1\); in order for this equation to also be valid for the boundary atoms \(n=1,N\), what (somewhat artificial) assumption does one need to make about the topology of the chain?

Solution #\(7\): For each \(n=2,3,…,N-2,N-1\), the longitudinal displacement satisfies Newton’s second law in the form:

\[m\ddot{x}_n=-k_s(x_n-x_{n-1})-k_s(x_n-x_{n+1})=-k_s(2x_n-x_{n-1}-x_{n+1})\]

If one were to use the same setup as in Problem #\(6\) where the boundary masses are simply connected to walls, or perhaps not connected to anything at all, then this equation of motion would not be valid for \(n=1,N\); however if one imposes an \(S^1\) topology on the chain by identifying \(x_n=x_m\pmod N\) (as if the \(N\) masses had been stringed along a circle instead of a line), then the equation becomes valid for \(n=1,N\) as well. It is for this simple convenient reason that periodic boundary conditions are commonly assumed; for \(N\gg 1\) it is intuitively plausible that the exact choice of boundary conditions shouldn’t affect bulk physics as bulk\(\cap\)boundary\(=\emptyset\).

Problem #\(8\): If \(\Delta x\) denotes the lattice spacing between adjacent masses \(m\) when the entire monatomic chain is at rest, show that the equation of motion coarse grains in the limit \(\Delta x\to 0\) to the \(1\)D free, dispersionless wave equation:

\[\frac{\partial^2\psi}{\partial (ct)^2}-\frac{\partial^2\psi}{\partial x^2}=0\]

where \(c^2=\frac{k_s}{m}(\Delta x)^2\), and the longitudinal displacement \(x_n\mapsto\psi\) has been promoted to a scalar field (and renamed to avoid confusion with the spatial coordinate \(x\)).

Solution #\(8\): One simply has to invoke the symmetric limit formulation of the second derivative:

\[\psi^{\prime\prime}(x)=\lim_{\Delta x\to 0}\frac{\psi(x+\Delta x)+\psi(x-\Delta x)-2\psi(x)}{(\Delta x)^2}\]

Problem #\(9\): Motivated by the result of Solution #\(8\), what is a reasonable ansatz for the longitudinal displacement \(x_n(t)\)? What is the connection between this result and the earlier discussion of normal modes?

Solution #\(9\): Motivated by the usual plane wave solutions \(\psi(x,t)=e^{i(kx-\omega_k t)}\) (with linear dispersion relation \(\omega_k=ck\)) that form the usual Fourier basis for arbitrary solutions to the wave equation, one can similarly write \(x_n(t)=e^{i(kn\Delta x-\omega_k t)}\), where \(x\mapsto n\Delta x\) is the analog of the spatial coordinate. Plugging this into the equation of motion, one finds instead that e.g. a sound wavepacket will disperse as it propagates through this monatomic chain:

\[\omega_k=2\sqrt{\frac{k_s}{m}}\biggr|\sin\frac{k\Delta x}{2}\biggr|\]

which can be plotted in the first Brillouin zone \(k\Delta x\in[-\pi,\pi)\):

Furthermore, the low-\(|k|\) limit of the group (or in this case phase) velocity \(c=\frac{\partial\omega}{\partial k}\) agrees with the linear dispersion relation from Problem #\(8\); this makes sense because the wave equation was obtained by coarse graining which is equivalent to assuming \(|k|\Delta x\ll 1\).

In general, whenever one finds a linear dispersion relation \(\omega_k\sim k\), it suggests massless particles (e.g. photons, phonons); this is to be contrasted with quadratic dispersion relations \(\omega_k\sim k^2\) which suggest massive particles (e.g. electrons in a tight-binding lattice).

Finally, to connect back to normal modes, because one has \(N\gg 1\) atoms coupled harmonically in the monatomic chain, one of course expects the system to have \(N\gg 1\) normal modes. And indeed the ansatz \(x_n(t)=e^{i(kn\Delta x-\omega_k t)}\) is exactly the form of a normal mode in which all atoms oscillate at the same frequency \(\omega_k\) and (on symmetry grounds) with the same amplitude, but where there is the same constant phase offset of \(x_{n+1}(t)/x_n(t)=e^{ik\Delta x}\) between each pair of adjacent atoms in the chain (again, this phase shift is the same between all pairs of atoms on symmetry grounds). Hopefully this gives the vibe of a sound wave propagating through the chain.

It’s as if we guessed ahead of time the eigenvectors \(x_n(t)\) of the matrix \(\omega^2\) in Solution #\(2\) and then simply acted with \(\omega^2\) on the ansatz eigenvector \( x_n(t)\) and checked that it scales by an eigenvalue \(\omega_k^2\) which gave the dispersion relation (again, usually one finds eigenvalues of a matrix before eigenvectors! This was only possible because of the symmetry; in general, circulant matrices are always diagonalized by the discrete Fourier transform).

Finally, as a minor aside, the earlier choice of periodic boundary condition \(x_{n}(t)\equiv x_{m}(t)\pmod N\) forces \(e^{ikN\Delta x}=1\) which quantizes the wavenumber \(k_m=\frac{2\pi m}{N\Delta x}\) for \(m\in\textbf Z\) (not to be confused with the mass \(m\) obviously); within the first Brillouin zone, there are \(N\) values of \(m=-N/2,…,N/2\) which are the \(N\) normal modes of the chain (i.e. each \(m\to k_m\to\omega_{k_m}\to\omega^2_{k_m}\) corresponding to one of the \(N\) eigenvalues of the \(N\times N\) \(\omega^2\) matrix).

So technically, for a finite monatomic chain of \(N\) atoms, there are only \(N\) combinations of \((k,\omega_k)\) on the dispersion graph which represent permissible normal modes, but as \(N\to\infty\) this quantization will become invisible with \(k,\omega_k\in\textbf R\).

Problem #\(10\): Repeat Problem #\(9\) but for a diatomic chain of masses. Hence show that there are \(2\) qualitatively different classes of normal modes that can be excited in this system, called acoustic modes and optical modes with respective dispersion relations:

\[\omega_{\pm}^2=\frac{k_s}{\mu}\left(1\pm\sqrt{1-\frac{4\mu\sin^2 k\Delta x}{m+M}}\right)\]

with \(k\Delta x\in[-\pi/2,\pi/2)\).

Solution #\(10\): On symmetry grounds, any normal mode would be expected to be of the form:

\[x_{2n}(t):=\alpha e^{i(k2n\Delta x-\omega_k t)}\]

\[x_{2n+1}(t):=\beta e^{i(k(2n+1)\Delta x-\omega_k t)}\]

where \(\alpha,\beta\in\textbf C\). Substituting into the equation of motion, one has:

\[\begin{pmatrix}2k_s-m\omega_k^2&-2k_s\cos k\Delta x\\ -2k_s\cos k\Delta x& 2k_s-M\omega_k^2\end{pmatrix}\begin{pmatrix}\alpha\\\beta\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\]

in order to get a non-trivial normal mode \((\alpha,\beta)\neq (0,0)\), require the determinant to vanish which gives the desired result (it is instructive to check that the quantity under the square root is indeed non-negative as required by Hermiticity of the matrix).

It is instructive to look at how the diatomic chain reduces to the monatomic chain by setting \(m=M=2\mu\). Then the optical branch becomes:

\[\omega_+=2\sqrt{\frac{k_s}{m}}\biggr|\cos\frac{k\Delta x}{2}\biggr|\]

while the acoustic branch reduces to the single branch that one had earlier for the monatomic chain:

\[\omega_-=2\sqrt{\frac{k_s}{m}}\biggr|\sin\frac{k\Delta x}{2}\biggr|\]

Nevertheless, in a monatomic chain there are no optical phonons, only acoustic phonons because it is \(\Delta x\)-periodic so the Brillouin zone ranges from \(-\pi\leq k\Delta x<\pi\) rather than \(-\pi/2\leq k\Delta x<\pi/2\) as appropriate for a \(2\Delta x\)-periodic diatomic chain (remember, there must still be \(N\) normal modes!). Nevertheless, this does provide intuition for why there is an optical branch (and this procedure is also generalizable to triatomic, etc. chains):

i.e. because the Brillouin zone of the diatomic chain is half as wide as the monatomic chain, so in the reduced Brillouin zone scheme for the diatomic chain, the parts from the monatomic chain are reflected inward, giving the \(\sim\cos\) shape of the optical branch (note these branches are very much analogous to a band structure for phonons instead of electrons; indeed because in a generic diatomic chain \(m\neq M\), this is the origin of the “branch gap” \((\omega_+-\omega_-)_{k=\pm\pi/2\Delta x}=\sqrt{2k_s}|m^{-1/2}-M^{-1/2}|\) between the acoustic and optical branches of the phonon spectrum \(\omega_k\) at the boundary of the Brillouin zone, cf. band gaps. In particular, \((\omega_+-\omega_-)_{k=\pm\pi/2\Delta x}\to 0\) as \(m\to M\)).

Back to the diatomic chain, for \(|k|\Delta x\ll 1\):

\[\omega_+\approx\sqrt{\frac{2k_s}{\mu}}\left(1-\frac{\mu}{2(m+M)}k^2\Delta x^2\right)\to\sqrt{\frac{2k_s}{\mu}}\]

\[\omega_-\approx\sqrt{\frac{2k_s}{m+M}}|k|\Delta x\to 0\]

and the group velocities are:

\[\frac{\partial\omega_{\pm}}{\partial k}=\frac{k_s^2\Delta x\sin 2k\Delta x}{\omega_{\pm}(\mu\omega_{\pm}^2-k_s)(m+M)}\]

and flatten to zero \(\partial\omega_{\pm}/\partial k\to 0\) at the boundary \(k\to\pm\pi/2\Delta x\) of the Brillouin zone, just as one finds in band structure (e.g. for nearly free electrons). All this is summarized in this refined picture:

Problem #\(11\): How do the atoms in the diatomic chain actually move in acoustic vs. optical normal modes?

Solution #\(11\): In general, the phasor ratios of \(m\) vs. \(M\) atom displacements (viewed as functions of \(k\)) are:

\[\left(\frac{\beta}{\alpha}\right)_{\pm}=\left(1-\frac{m\omega_{\pm}^2}{2k_s}\right)\sec k\Delta x\]

There are \(2\) interesting limits of this expression, namely \(k=0\) and \(k=\pm\pi/2\Delta x\). In the former case:

\[\left(\frac{\beta}{\alpha}\right)_{+}(k=0)=-\frac{m}{M}\]

\[\left(\frac{\beta}{\alpha}\right)_{-}(k=0)=1\]

So at low-\(k\), the diatomic chain in an acoustic normal mode looks like neighbouring atoms oscillating in phase with equal amplitude whereas the optical mode looks like a \(\pi\)-radian out-of-phase oscillation with the lighter atoms oscillating more than the heavier atoms by the mass ratio \(M/m\), as intuitively reasonable. This also explains the name “optical branch”; if the atoms in the diatomic chain are in fact ions of opposite charge as in an \(\text{NaCl}\) crystal, then one has pairs of oscillating electric dipoles which can couple to light.

Meanwhile, at the boundary of the Brillouin zone, one finds:

\[\left(\frac{\beta}{\alpha}\right)_{\pm}(k=\pm\pi/2\Delta x)=\frac{M-m\mp|M-m|}{2M}\sec\pm\frac{\pi}{2}\]

where although \(\sec\pm\pi/2=\infty\) diverges, causing \((\beta/\alpha)_-=\infty\), actually the factor in front goes to zero even faster than the secant diverges as \(k\to\pm\pi/2\Delta x\) so that actually \((\beta/\alpha)_+=0\). This is a (longitudinal) standing wave! i.e. in the optical mode, one has nodes at the heavier atoms \(M\) while the light atoms represent displacement antinodes, and vice versa in the acoustic mode (intuitively, \(k=\pi/2\Delta x\Leftrightarrow\lambda=4\Delta x\)).

Notice the group velocity \((\partial\omega/\partial k)_{k=\pm\pi/2\Delta x}\) also tends to zero at the Brillouin zone boundary…that’s just another way of saying it’s a standing wave!

Another useful intuition is that, say near \(k=0\), the acoustic and optical normal modes look the way they do such that atoms can be paired up at any moment in time with equal and opposite momenta, giving the overall collective excitation/phonon also zero momentum! So there is a direct connection between \(mv\) momentum and \(\hbar k\) momentum (ACTUALLY ARE WE SURE THIS IS TRUE?).

Problem #\(12\): Just to emphasize again: on the one hand, one can generically refer to \(\omega_k\) as a dispersion relation for a particular kind of wave; here what is a better name for \(\omega_k\)?

Solution #\(12\): The phrase “normal mode spectrum” is useful to keep in mind.

The purpose of this post is to answer some questions posed to me by ChatGPT regarding my understanding of the \(z\)-transform in digital signal processing and mathematics more broadly. My inquiry to it was simply: “Ask me some questions to test my understanding of the \(z\)-transform”. Here were ChatGPT’s questions, followed by my answers.

Question #1: What is the general definition of the \(z\)-transform for a discrete-time signal \(x_n\)?

My Answer: Given a sequence \((x_n)_{n=-\infty}^{\infty}\) of (possibly complex) numbers, the \(z\)-transform \(X(z)\) of the sequence \((x_n)_{n\in\textbf Z}\) is a map \(z\in\textbf C\to X(z)\in\textbf C\) on the complex plane \(\textbf C\) defined by the formal Laurent series about the origin:

\[X(z):=\sum_{n\in\textbf Z}x_n z^{-n}\]

It is also denoted \(X(z)=\mathcal Z\{x\}(z)\). More precisely, this is the bilateral \(z\)-transform and one can also consider sequences \((x_n)_{n=0}^{\infty}\) defined only for non-negative \(n\geq 0\) along with the unilateral \(z\)-transform \(X(z):=\sum_{n=0}^{\infty}x_n z^{-n}\). This then resembles the form of an (ordinary) generating function for the sequence \((x_n)_{n=0}^\infty\).

Question #2: For the discrete unit impulse signal \(x_n=\delta_{n,0}\), what is the z-transform?

My Answer: As an Einstein series over \(n\in\textbf Z\), we have \(X(z)=\delta_{n,0}z^{-n}\) which contracts to \(X(z)=z^{-0}=1\).

Question #3: What does the region of convergence (ROC) in the \(z\)-transform tell you about a system or signal?

My Answer: The region of convergence (ROC), perhaps more descriptively called the annulus of convergence, is simply the set of all \(z\in\textbf C\) for which the formal Laurent series defining the \(z\)-transform is actually a convergent infinite series. It provides information about both stability and causality.

Question #4: How do you determine if a discrete-time dynamical system is stable using the \(z\)-transform?

My Answer: Recalling that a discrete-time dynamical system \((x_n)_{n=-\infty}^{\infty}\) is said to be stable (or asymptotically stable) iff \(\lim_{n\to\infty}x_n=0\), the sufficient and necessary criterion for stability is that the poles of its \(z\)-transform \(X(z)\) must all lie in the interior of \(U(1)\) i.e. \(|z_{\text{poles}}|<1\).

Question #5: Compute the \(z\)-transform of \(x_n=\left(-\frac{1}{2}\right)^nu_n\) where \(u_n\) is the unit step function \(u_n:=[n\geq 0]\).

My Answer:

\[X(z)=\sum_{n=-\infty}^{\infty}\left(-\frac{1}{2}\right)^nu_nz^{-n}=\sum_{n=0}^{\infty}\left(-\frac{1}{2z}\right)^n=\frac{1}{1+1/2z}\]

where the annulus of convergence is \(|z|>1/2\).

Question #6: Explain how the \(z\)-transform simplifies the process of convolution between two discrete-time signals \(x_n,y_n\).

My Answer: The \(z\)-transform of the discrete convolution \((x*y)_n\) of two discrete-time signals \(x_n,y_n\) is the product of their individual \(z\)-transforms. This is one version of the convolution theorem:

\[\mathcal Z\{x*y\}(z)=\mathcal Z\{x\}(z)\mathcal Z\{y\}(z)\]

Put another way, to compute the convolution \((x*y)_n\) of two discrete-time signals, one can instead compute \((x*y)_n=\mathcal Z^{-1}(\mathcal Z\{x\}(z)\mathcal Z\{y\}(z))_n\).

Question #7: For the \(z\)-transform \(X(z)=\frac{z+1}{z^2-3z/4+1/8}\), what is the corresponding time-domain signal \(x_n\)?

My Answer: The \(z\)-transform can first be simplified using partial fractions:

\[X(z)=\frac{z+1}{z^2-3z/4+1/8}=\frac{z+1}{(z-1/2)(z-1/4)}=\frac{6}{z-1/2}-\frac{5}{z-1/4}\]

Using the standard complex analysis, one can check that the inverse \(z\)-transform is given by the contour integral:

\[x_n=\frac{1}{2\pi i}\oint_{\Gamma}X(z)z^{n-1}dz\]

where \(\Gamma\) is any counterclockwise-oriented curve that has winding number \(\text{Wind}(\Gamma,0)=1\) around the origin \(0\in\textbf C\) and lies within the annulus of convergence of the \(z\)-transform \(X(z)\) (which in this case is just \(\textbf C-\{1/2,1/4\}\)). Thus, the inverse \(z\)-transform yields:

\[x_n=6(1/2)^{n-1}-5(1/4)^{n-1}\]

Question #8: Explain how the \(z\)-transform is related to the discrete-time Fourier transform (DTFT).

The discrete time Fourier transform of a sequence \((x_n)_{n\in\textbf Z}\) is simply its \(z\)-transform \(X(z)\) but evaluated on \(U(1)\) i.e. \(z=e^{i\omega}\). Thus, the DTFT is:

\[X(e^{i\omega})=\sum_{n\in\textbf Z}x_ne^{-i\omega n}\]