Chapter 31 Exercises

28 problems spanning computation, analysis, and conceptual reasoning. Difficulty ratings: ★ (routine), ★★ (intermediate), ★★★ (challenging), ★★★★ (advanced/research-level).

Section 31.1–31.2: Derivation and Interpretation

Problem 31.1 ★ Starting from the time-evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$, show explicitly that inserting one completeness relation $\hat{I} = \int dx_1\,|x_1\rangle\langle x_1|$ gives:

$$K(x_f, t; x_i, 0) = \int dx_1\, K(x_f, t; x_1, t/2)\, K(x_1, t/2; x_i, 0)$$

Interpret this physically as the composition of two propagations through the intermediate point $x_1$.

Problem 31.2 ★ The Trotter decomposition states $e^{-i(\hat{A}+\hat{B})\epsilon} \approx e^{-i\hat{A}\epsilon}\,e^{-i\hat{B}\epsilon} + O(\epsilon^2)$. (a) For $\hat{A} = \hat{p}^2/(2m\hbar)$ and $\hat{B} = V(\hat{x})/\hbar$, compute the commutator $[\hat{A},\hat{B}]$ and show the leading correction is proportional to $\epsilon^2$. (b) Show that the symmetric Trotter decomposition $e^{-i\hat{A}\epsilon/2}\,e^{-i\hat{B}\epsilon}\,e^{-i\hat{A}\epsilon/2}$ has errors of order $O(\epsilon^3)$. Why is this useful for numerical implementations?

Problem 31.3 ★★ Starting from the discretized path integral, verify explicitly for $N = 2$ time slices that the free-particle propagator gives the correct result $K_{\text{free}} = \sqrt{m/(2\pi i\hbar t)}\,\exp[im(x_f - x_i)^2/(2\hbar t)]$. That is, show:

$$\left(\frac{m}{2\pi i\hbar\epsilon}\right) \int_{-\infty}^{\infty} dx_1\, \exp\left\{\frac{im}{2\hbar\epsilon}\left[(x_1 - x_i)^2 + (x_f - x_1)^2\right]\right\} = \sqrt{\frac{m}{2\pi i\hbar(2\epsilon)}}\, e^{im(x_f - x_i)^2/(2\hbar\cdot 2\epsilon)}$$

Problem 31.4 ★★ Consider a particle on a ring of circumference $L$ (periodic boundary conditions: $x \equiv x + L$). The path integral sums over paths from $x_i$ to $x_f$, but now paths can wind around the ring $n$ times. (a) Argue that the propagator is $K(x_f, t; x_i, 0) = \sum_{n=-\infty}^{\infty} K_{\text{free}}(x_f + nL, t; x_i, 0)$. (b) Show that the Poisson summation formula converts this into the spectral decomposition $K = (1/L)\sum_k e^{ikx_f}e^{-ikx_i}e^{-i\hbar k^2 t/(2m)}$ with $k = 2\pi n/L$. (c) Read off the energy eigenvalues and verify they match the quantum mechanics of a particle on a ring.

Problem 31.5 ★★ A particle in one dimension has the Hamiltonian $\hat{H} = \hat{p}^2/(2m) + V(\hat{x})$. Show that the path integral measure can equivalently be written using both position and momentum integrals (the phase-space path integral):

$$K = \int \mathcal{D}[x]\mathcal{D}[p]\, \exp\left\{\frac{i}{\hbar}\int_0^t \left[p\dot{x} - H(p,x)\right]dt'\right\}$$

where $H(p,x) = p^2/(2m) + V(x)$ is the classical Hamiltonian. Why does integrating out $p$ recover the Lagrangian path integral?

Problem 31.6 ★★★ Explain, using the path integral, why the Aharonov-Bohm effect (a charged particle encircling a solenoid) exists. Specifically: (a) Show that a magnetic vector potential $\mathbf{A}$ modifies the path integral weight by a phase factor $\exp\left[\frac{iq}{\hbar c}\int \mathbf{A}\cdot d\mathbf{l}\right]$ along each path. (b) For two paths encircling a solenoid on opposite sides, show the relative phase is $q\Phi/(c\hbar)$ where $\Phi$ is the magnetic flux. (c) Explain why the interference pattern depends on the enclosed flux even though $\mathbf{B} = 0$ everywhere the particle travels.

Section 31.3: Free-Particle Propagator

Problem 31.7 ★ Verify directly that the free-particle propagator satisfies the initial condition $\lim_{t\to 0^+} K_{\text{free}}(x_f, t; x_i, 0) = \delta(x_f - x_i)$ by showing: (a) As $t \to 0^+$, the Gaussian becomes infinitely narrow. (b) The integral $\int K_{\text{free}}(x_f, t; x_i, 0)\, dx_f = 1$ for all $t > 0$.

Problem 31.8 ★★ Using the free-particle propagator, compute $\Psi(x, t)$ for the initial condition:

$$\Psi(x, 0) = \left(\frac{1}{2\pi\sigma_0^2}\right)^{1/4} \exp\left(-\frac{x^2}{4\sigma_0^2} + ik_0 x\right)$$

(a Gaussian wave packet with initial momentum $\hbar k_0$). Show that:

$$|\Psi(x,t)|^2 = \frac{1}{\sqrt{2\pi}\sigma(t)}\exp\left[-\frac{(x - \hbar k_0 t/m)^2}{2\sigma(t)^2}\right]$$

with $\sigma(t) = \sigma_0\sqrt{1 + \hbar^2 t^2/(4m^2\sigma_0^4)}$, reproducing the result of Chapter 7.

Problem 31.9 ★★ The free-particle propagator in $d$ dimensions is:

$$K_{\text{free}}(\mathbf{r}_f, t; \mathbf{r}_i, 0) = \left(\frac{m}{2\pi i\hbar t}\right)^{d/2} \exp\left(\frac{im|\mathbf{r}_f - \mathbf{r}_i|^2}{2\hbar t}\right)$$

(a) Verify this by separation of variables (the $d$-dimensional propagator is a product of $d$ one-dimensional propagators). (b) For $d = 3$, show that this reduces to the known Green's function for the free Schrödinger equation.

Problem 31.10 ★★★ Consider a free particle in the presence of a constant external force $F$ (potential $V(x) = -Fx$). (a) Find the classical trajectory from $x_i$ to $x_f$ in time $t$. (b) Compute the classical action $S_{\text{cl}}$. (c) Show that the action is exactly quadratic in $x_i$ and $x_f$, so the path integral is Gaussian and the semiclassical result is exact. (d) Write down the exact propagator and verify it satisfies the Schrödinger equation.

Section 31.4: Stationary Phase and the Classical Limit

Problem 31.11 ★ The one-dimensional stationary phase approximation states:

$$\int_{-\infty}^{\infty} g(x)\, e^{i\lambda f(x)}\, dx \approx g(x_0)\sqrt{\frac{2\pi}{\lambda|f''(x_0)|}}\, e^{i\lambda f(x_0) \pm i\pi/4}$$

for large $\lambda$, where $f'(x_0) = 0$ and the $\pm$ corresponds to $f''(x_0) \gtrless 0$.

Apply this to evaluate $\int_{-\infty}^{\infty} e^{i\lambda(x^3/3 - x)}\, dx$ for large $\lambda$. Find the stationary phase points and the leading approximation.

Problem 31.12 ★★ A particle of mass $m$ moves in the potential $V(x) = mgx$ (uniform gravitational field). (a) Find the classical path from $(x_i, 0)$ to $(x_f, t)$. (b) Compute $S_{\text{cl}}$ and verify that $\partial S_{\text{cl}}/\partial x_f = p_f$ (the final momentum) and $-\partial S_{\text{cl}}/\partial x_i = p_i$ (the initial momentum) — the Hamilton-Jacobi relations. (c) Show that $-\partial S_{\text{cl}}/\partial t = E$ (the energy).

Problem 31.13 ★★ Derive the WKB wave function $\psi(x) \propto |p(x)|^{-1/2}\,e^{\pm(i/\hbar)\int^x p(x')\,dx'}$ from the path integral by applying the stationary phase approximation to the energy-domain propagator $G(x_f, x_i; E) = \int_0^{\infty} K(x_f, t; x_i, 0)\, e^{iEt/\hbar}\, dt$.

Problem 31.14 ★★★ A charged particle of mass $m$ and charge $q$ moves in a uniform magnetic field $\mathbf{B} = B\hat{z}$ in two dimensions. The Lagrangian is $L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2) + \frac{qB}{2c}(x\dot{y} - y\dot{x})$ (symmetric gauge). (a) Show that the classical trajectories are circles (cyclotron orbits). (b) Since $L$ is quadratic, the path integral is exact. Using the result from the QHO (with appropriate modifications), show that the propagator gives the Landau level energies $E_n = \hbar\omega_c(n + 1/2)$ where $\omega_c = qB/(mc)$.

Section 31.5: QHO Path Integral

Problem 31.15 ★ Verify that the classical solution $x_{\text{cl}}(t') = [x_i \sin\omega(t - t') + x_f\sin(\omega t')]/\sin(\omega t)$ satisfies: (a) The equation of motion $\ddot{x}_{\text{cl}} + \omega^2 x_{\text{cl}} = 0$ (b) The boundary conditions $x_{\text{cl}}(0) = x_i$ and $x_{\text{cl}}(t) = x_f$

Problem 31.16 ★★ Show that the QHO classical action reduces to the free-particle action in the limit $\omega \to 0$:

$$\frac{m\omega}{2\sin(\omega t)}\left[(x_i^2 + x_f^2)\cos(\omega t) - 2x_i x_f\right] \xrightarrow{\omega\to 0} \frac{m(x_f - x_i)^2}{2t}$$

by expanding $\sin(\omega t)$ and $\cos(\omega t)$ to the appropriate orders in $\omega$.

Problem 31.17 ★★ From the QHO propagator, extract the ground-state wave function. (a) Take the limit $t \to \infty$ (more precisely, $t \to -i\infty$, i.e., large imaginary time) in the propagator. Show that the sum over energy eigenstates is dominated by the ground state. (b) Read off $\psi_0(x_f)\psi_0^*(x_i) = \lim_{T\to\infty} e^{iE_0 T/\hbar} K(x_f, T; x_i, 0)$ (with $T \to -i\infty$). (c) Verify that you obtain $\psi_0(x) = (m\omega/(\pi\hbar))^{1/4}\exp[-m\omega x^2/(2\hbar)]$.

Problem 31.18 ★★★ At the half-period $t = \pi/\omega$, the QHO propagator becomes $K(x_f, \pi/\omega; x_i, 0) \propto \delta(x_f + x_i)$. (a) Prove this by taking the limit $\omega t \to \pi$ in the exact propagator. (b) What does this mean physically? If the initial state is $\Psi(x, 0) = f(x)$, what is $\Psi(x, \pi/\omega)$? (c) Connect this to the parity operator $\hat{\Pi}$: show that $\hat{U}(\pi/\omega) = e^{-i\pi/2}\hat{\Pi}$ (up to a phase).

Problem 31.19 ★★★★ The Mehler kernel: Show that the QHO propagator can be written as a sum over energy eigenstates:

$$K_{\text{QHO}}(x_f, t; x_i, 0) = \sum_{n=0}^{\infty} \psi_n(x_f)\psi_n^*(x_i)\, e^{-i\omega(n+1/2)t}$$

by verifying that the bilinear generating function for Hermite polynomials (Mehler's formula):

$$\sum_{n=0}^{\infty} \frac{H_n(u)H_n(v)}{2^n n!}\, r^n = \frac{1}{\sqrt{1-r^2}}\exp\left[\frac{2uvr - (u^2+v^2)r^2}{1-r^2}\right]$$

with $r = e^{-i\omega t}$ reproduces the exact propagator derived in the text.

Section 31.6: Statistical Mechanics Connection

Problem 31.20 ★ Perform the Wick rotation $t \to -i\tau$ on the free-particle propagator $K_{\text{free}} = \sqrt{m/(2\pi i\hbar t)}\exp[im(x_f-x_i)^2/(2\hbar t)]$ and show: (a) The result is a normalized Gaussian in $x_f - x_i$ (a real, positive probability distribution). (b) Identify the diffusion constant $D$ such that the result matches the heat kernel $K_{\text{diffusion}} = (4\pi D\tau)^{-1/2}\exp[-(x_f-x_i)^2/(4D\tau)]$.

Problem 31.21 ★★ Compute the QHO partition function $Z = 1/[2\sinh(\beta\hbar\omega/2)]$ by two methods: (a) Direct summation $Z = \sum_{n=0}^{\infty} e^{-\beta E_n}$ using $E_n = \hbar\omega(n + 1/2)$. (b) The Wick-rotated path integral trace $Z = \int dx\, K(x, -i\beta\hbar; x, 0)$ using the exact QHO propagator. Verify that both methods agree.

Problem 31.22 ★★ From the QHO partition function, compute: (a) The average energy $\langle E \rangle = -\partial\ln Z/\partial\beta$. (b) The heat capacity $C = \partial\langle E\rangle/\partial T$. (c) Show that $\langle E \rangle \to k_B T$ at high temperature (classical equipartition) and $\langle E \rangle \to \hbar\omega/2$ at low temperature (zero-point energy). (d) Plot $\langle E\rangle/(k_B T)$ vs. $k_B T/(\hbar\omega)$ and identify the crossover between quantum and classical regimes.

Problem 31.23 ★★★ The thermal density matrix for the QHO can be written as a path integral:

$$\rho(x_f, x_i; \beta) = \frac{1}{Z} K(x_f, -i\beta\hbar; x_i, 0)$$

(a) Write this explicitly using the QHO propagator with the Wick rotation. (b) Compute the diagonal element $\rho(x, x; \beta)$ and show it gives the probability distribution of position at temperature $T$. (c) In the limit $T \to 0$, show $\rho(x, x; 0) = |\psi_0(x)|^2$. (d) In the limit $T \to \infty$, show $\rho(x, x; \infty) \propto e^{-m\omega^2 x^2/(2k_B T)}$ (Boltzmann distribution).

Problem 31.24 ★★★ Consider a double-well potential $V(x) = \lambda(x^2 - a^2)^2$. (a) Find the instanton solution $x_{\text{inst}}(\tau)$ that interpolates from $x = -a$ to $x = +a$ in Euclidean time. (b) Compute the Euclidean action $S_E$ of the instanton. (c) The tunnel splitting between the ground and first excited state is $\Delta E \propto e^{-S_E/\hbar}$. Explain qualitatively why this is nonperturbative — invisible to any finite-order Taylor expansion in $\lambda$.

Section 31.7: Feynman Diagrams Preview

Problem 31.25 ★★ Wick's theorem for a one-dimensional Gaussian integral: Let $\langle f(x) \rangle = \int_{-\infty}^{\infty} f(x)\, e^{-\alpha x^2/2}\, dx\, /\, \int_{-\infty}^{\infty} e^{-\alpha x^2/2}\, dx$. (a) Compute $\langle x^2 \rangle$, $\langle x^4 \rangle$, and $\langle x^6 \rangle$. (b) Verify that $\langle x^{2n}\rangle = (2n-1)!!\,\langle x^2\rangle^n$ (the product of all pairings). (c) Show that $\langle x^{2n+1}\rangle = 0$ for all $n$.

Problem 31.26 ★★ Consider the anharmonic oscillator with Lagrangian $L = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2 - \lambda x^4$. (a) Write the first-order correction to the propagator in terms of the free QHO Green's function $G(t_1, t_2)$. (b) How many distinct diagrams contribute? Draw them. (c) Which diagrams are connected and which are disconnected?

Problem 31.27 ★★★ The generating functional for the QHO with an external source $J(t')$ is:

$$Z[J] = \int \mathcal{D}[x]\, \exp\left\{\frac{i}{\hbar}\int_0^t \left[\frac{1}{2}m\dot{x}^2 - \frac{1}{2}m\omega^2 x^2 + J(t')x(t')\right]dt'\right\}$$

(a) Show that this is a Gaussian integral and evaluate it in terms of the Green's function $G(t_1, t_2)$ (the inverse of the operator $-m\partial_t^2 - m\omega^2$). (b) Show that $\langle x(t_1)x(t_2)\rangle = (\hbar/i)^2 \delta^2 \ln Z[J]/\delta J(t_1)\delta J(t_2)\big|_{J=0}$. (c) This is the path integral version of the time-ordered correlation function $\langle 0|\hat{T}\{\hat{x}_H(t_1)\hat{x}_H(t_2)\}|0\rangle$. Explain the connection.

Problem 31.28 ★★★★ The sign of the prefactor and Maslov indices. When the classical path passes through a focal point (where $\partial^2 S_{\text{cl}}/\partial x_f \partial x_i = 0$), the semiclassical prefactor changes sign and the propagator acquires an extra phase of $-\pi/2$. (a) Show that for the QHO, focal points occur at $t = n\pi/\omega$ for integer $n$. (b) Verify that the exact QHO propagator picks up a phase $e^{-i\pi/2}$ each time $t$ crosses a focal point (compare the propagator just before and just after $t = \pi/\omega$). (c) This phase is the Maslov index. Explain its connection to the Morse index (number of conjugate points along the classical trajectory) and the metaplectic representation.