Author Archives: wdengquantum.me

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 →

The purpose of this post is to lay out the basic theory of ensembles in statistical mechanics in addition to some examples. At the end, the goal will be to convincingly demonstrate ensemble equivalence. Any system (e.g. a gas, a … Continue reading →

Problem: What is the state space \(\mathcal H_{\text{TB}}\) of the tight-binding model on a lattice \(\Lambda\) in \(\mathbf R^d\), ignoring spin degrees of freedom and assuming a single state/”orbital” per lattice point in \(\Lambda\). Solution: The tight-binding model assumes a … Continue reading →

Problem: Consider a classical particle subject to Newton’s second law \(\dot{\textbf p}=\textbf F\). If one takes the dot product of both sides with the particle’s velocity \(\dot{\textbf x}\) so that \(\dot{\textbf p}\cdot\dot{\textbf x}=\textbf F\cdot\dot{\textbf x}\), the left hand side is … Continue reading →

The purpose of this post is to contrast the classical treatment of Rutherford scattering with its quantum mechanical treatment. At the end, the surprise will be that for certain important quantities such as the differential cross-section, the classical and quantum … Continue reading →

Problem: Mobius transformations (also called fractional linear transformations) are maps \(\mathcal M:\textbf C\cup\{\infty\}\to\textbf C\cup\{\infty\}\) on the Riemann sphere \(\textbf C\cup\{\infty\}\) of the form \(\mathcal M(z):=\frac{az+b}{cz+d}\) for \(\det\begin{pmatrix}a&b\\c&d\end{pmatrix}\neq 0\). The purpose of this problem is to gain a deeper appreciation for … Continue reading →

The purpose of this post is to prove a simple but somewhat unintuitive result in nonrelativistic \(1\)D scattering of quantum particles. Specifically, consider a potential energy landscape \(V(x)\) that decays asymptotically to \(\lim_{|x|\to\infty}V(x)=0\) at \(|x|\to\infty\) which may be asymmetric so … Continue reading →

Given a mixed ensemble \(\{(|\psi_n\rangle, p_n)\}\) of pure quantum states \(|\psi_n\rangle\in\mathcal H\) with statistical probabilities \(p_n\in[0,1]\), the Hermitian, positive semi-definite, unit trace density operator \(\rho_{\{(|\psi_n\rangle, p_n)\}}:\mathcal H\to\mathcal H\) of that mixed ensemble is defined by the formula: \[\rho_{\{(|\psi_n\rangle, p_n)\}}:=\sum_np_n|\psi_n\rangle\langle\psi_n|\] In … Continue reading →