Lecture Notes#

Homework#

1. Microcanonical from max entropy. The microcanonical ensemble assigns equal probability to all states with energy in a thin shell \([E, E+\delta E]\). Apply max-entropy reasoning to derive it.

(a) Maximize \(S = -k_B\sum_n p_n\,\ln p_n\) subject only to \(\sum_n p_n = 1\), where the sum runs over states inside the shell. Show that \(p_n = 1/\Omega(E)\) is uniform, where \(\Omega(E)\) is the number of states in the shell.

(b) Compute \(S = k_B\ln\Omega(E)\) — Boltzmann’s entropy formula.

(c) The canonical (Boltzmann) distribution \(p_n = \mathrm{e}^{-\beta E_n}/Z\) instead arises by adding the energy constraint \(\sum_n p_n E_n = U\). Without computing the full max-entropy stationary point, explain physically why dropping the energy constraint is unphysical for a system in contact with a thermal bath — i.e., why the bath’s role is to fix the mean energy, not the total state count.

2. Partition function derivatives. A two-level system has \(\hat{H} = E_0\vert 0\rangle\langle 0\vert + E_1\vert 1\rangle\langle 1\vert\) with \(E_0 < E_1\) and gap \(\Delta = E_1 - E_0\).

(a) Compute \(Z(\beta)\) explicitly.

(b) From (118), compute \(\langle E\rangle(T)\) and the heat capacity \(C(T)\). Sketch \(C(T)\) and identify the Schottky peak temperature \(k_B T \sim \Delta\).

(c) Verify the limits \(\langle E\rangle\to E_0\) as \(T\to 0\) and \(\langle E\rangle\to (E_0+E_1)/2\) as \(T\to\infty\).

3. Spectral sum equals trace. For the qubit of HW 3.4.1.2, verify \(Z(\beta) = \mathrm{Tr}\,\mathrm{e}^{-\beta\hat{H}}\) in two ways: (a) directly from the definition \(\sum_n\mathrm{e}^{-\beta E_n}\), and (b) by computing the matrix exponential \(\mathrm{e}^{-\beta\hat{H}}\) in the \(\{\vert 0\rangle,\vert 1\rangle\}\) basis and tracing. Confirm the two results agree.

4. Free-particle Wick rotation. For a free particle (\(V = 0\)) with real-time action \(S = \int_0^T\tfrac{1}{2}m\dot{x}^2\,\mathrm{d}t\):

(a) Apply \(t \to -\mathrm{i}\tau\), \(\dot{x}\to\mathrm{i}\,x'(\tau)\), \(\mathrm{d}t\to -\mathrm{i}\,\mathrm{d}\tau\) and show \(S \to +\mathrm{i}\,S_E\) with \(S_E = \int_0^{T_E}\tfrac{1}{2}m\,x'(\tau)^2\,\mathrm{d}\tau \ge 0\).

(b) Verify \(\mathrm{e}^{\mathrm{i}S/\hbar}\to\mathrm{e}^{-S_E/\hbar}\), and explain why this exponentially suppresses paths with large kinetic energy.

5. Harmonic oscillator Euclidean action. For \(V(x) = \tfrac{1}{2}m\omega^2 x^2\):

(a) Apply the Wick rotation to derive \(S_E = \int_0^{T_E}\bigl(\tfrac{1}{2}m\,x'(\tau)^2 + \tfrac{1}{2}m\omega^2 x^2\bigr)\,\mathrm{d}\tau\).

(b) Both terms in \(S_E\) are positive. Explain in one sentence why this guarantees convergence of the Euclidean path integral.

(c) For the trial path \(x(\tau) = A\cos(\omega\tau)\) with small \(A\), compute \(S_E\) over one period \(T_E = 2\pi/\omega\) and verify it is positive.

6. Time becomes temperature: numerical estimate. Estimate \(\hbar\beta\) for an electronic system at room temperature (\(T = 300\,\mathrm{K}\)).

(a) Compute \(\hbar\beta\) in seconds. How does this compare to the inverse Bohr-orbit frequency \(\hbar/E_{\mathrm{Ry}}\) where \(E_{\mathrm{Ry}} \approx 13.6\,\mathrm{eV}\)?

(b) On what physical timescale does Wick rotation matter for thermal physics? In one sentence, why is room-temperature electronic structure essentially “frozen” on the thermal-circle scale?

7. Free-particle thermal trace and equipartition. The Euclidean propagator of a free particle is \(K_E(x,\tau;x_0,0) = \sqrt{m/(2\pi\hbar\tau)}\,\exp[-m(x-x_0)^2/(2\hbar\tau)]\).

(a) Set \(\tau = \hbar\beta\), \(x = x_0\), and integrate over \(x_0\) in a box of length \(L\) to obtain \(Z_{\mathrm{free}}(\beta) = L/\lambda_{\mathrm{th}}\) with \(\lambda_{\mathrm{th}} = \sqrt{2\pi\hbar^2\beta/m}\).

(b) From \(\langle E\rangle = -\partial\ln Z/\partial\beta\), recover the 1D classical equipartition result \(\langle E\rangle = \tfrac{1}{2}k_B T\).

8. Thermal circle, geometrically. Sketch the imaginary-time circle of circumference \(\hbar\beta\) for two regimes: \(\hbar\beta \ll \lambda_{\mathrm{th}}/v\) (high \(T\)) and \(\hbar\beta \gg \lambda_{\mathrm{th}}/v\) (low \(T\)), where \(v = \sqrt{k_B T/m}\) is a typical thermal velocity.

(a) Argue that in the high-\(T\) limit a typical path barely moves during one trip around the circle, so the thermal trace reduces to a classical Boltzmann integral over a single position \(x_0\).

(b) Argue that in the low-\(T\) limit only the lowest-energy eigenstate contributes to \(Z(\beta)\). Show \(Z(\beta)\to \mathrm{e}^{-\beta E_0}\) as \(\beta\to\infty\) and conclude \(E_0 = -\lim_{\beta\to\infty}\beta^{-1}\ln Z(\beta)\) — the long-\(\beta\) projection onto the ground state.