Author Archives: wdengquantum.me

Problem: Write down a formula for the total energy \(E\) stored in the transverse and longitudinal acoustic phonons of an insulator at temperature \(T\). Solution: \[E=\int_0^{k_D}dk 3\frac{V}{(2\pi)^3}4\pi k^2\times\hbar\omega_k\times\frac{1}{e^{\beta\hbar\omega_k}-1}\] where the Debye wavenumber \(k_D\sim(N/V)^{1/3}\) is determined precisely by requiring \(\int_0^{k_D}dk 3\frac{V}{(2\pi)^3}4\pi … Continue reading →

Problem: What is the position space wavefunction \(\psi(\textbf x)\) of a photon in a box of volume \(V=L^3\) whose opposite faces are subject to periodic boundary conditions? What about reflecting boundary conditions? Solution: With periodic boundary conditions, the wavefunctions are … Continue reading →

The purpose of this post is to explore the rich physics encoded in the Van der Waals equation of state: \[\left(p+\frac{aN^2}{V^2}\right)(V-Nb)=NkT\] Essentially, the Van der Waals equation of state dispenses with two assumptions implicit in the ideal gas law: First, … Continue reading →

The purpose of this post is to demonstrate how the scattering operator \(S:\mathcal H_{\text{incident}}^{\infty}\to\mathcal H_{\text{scattered}}^{\infty}\), also called the \(S\)-operator for short, despite being defined to enact the scattering \(|\psi’_{\infty}\rangle=S|\psi_{\infty}\rangle\) of asymptotic incident waves \(|\psi_{\infty}\rangle\) off a potential \(V\) into asymptotic … Continue reading →

Problem: Explain what the “nearly” in “nearly free electrons” means precisely. Solution: It means the potential \(V\) experienced by the electrons is assumed to be weak in comparison to their kinetic energy \(V\ll T\) (in addition to being weak, it … Continue reading →

Problem: Before developing FGR, it is first necessary to lay out as a prerequisite the general framework of time-dependent perturbation theory (FGR is then a special case thereof). To this effect, consider the usual (Schrodinger picture) Hamiltonian decomposition \(H=H_0+V\), and … Continue reading →

The purpose of this post is to provide some examples of perturbation theory calculations in quantum mechanics by analyzing a specific perturbation called the Stark effect. For simplicity, we take as our quantum system a hydrogen atom with the usual … Continue reading →

Consider the three \(\ell=1\) spherical harmonics: \[Y_{1}^{-1}(\theta,\phi)=\sqrt{\frac{3}{8\pi}}\sin\theta e^{i\phi}\] \[Y_0^0(\theta)=\sqrt{\frac{3}{4\pi}}\cos\theta\] \[Y_{1}^{1}(\theta,\phi)=-\sqrt{\frac{3}{8\pi}}\sin\theta e^{-i\phi}\] Although the spherical harmonics are just functions of \(\theta,\phi\) on the sphere \(S^2\), one can introduce an \(r\)-dependence simply by multiplying each of them by \(r\). The resulting functions … Continue reading →

Problem: Working on the flat manifold of Minkowski spacetime \(X\cong\mathbf R^{1,3}\) equipped with the Lorentzian metric signature \((+,-,-,-)\), define what is meant by a Lorentz scalar, a Lorentz vector (also called a 4-vector), a Lorentz tensor, and a Lorentz spinor. … Continue reading →

Problem: Define an ideal gas (whether that be classical or quantum). Write down the single-particle canonical partition function \(Z_1\) for a classical, monatomic ideal gas, and hence the \(N\)-particle canonical partition function \(Z_N\) in terms of \(Z_1\) (assuming all \(N\) … Continue reading →