Chapter 31 Quiz

20 questions testing conceptual understanding and computational fluency. Answers in Appendix H.

Q1. The path integral formulation of quantum mechanics was developed by:

(a) Paul Dirac in 1930 (b) Richard Feynman in 1948 (c) Erwin Schrödinger in 1926 (d) Werner Heisenberg in 1925

Q2. In the path integral $K = \int \mathcal{D}[x]\, e^{iS[x]/\hbar}$, each path contributes with:

(a) A weight equal to its classical probability (b) A weight proportional to $e^{-S/\hbar}$ (exponentially suppressed for large action) (c) A complex phase $e^{iS/\hbar}$ of unit magnitude (d) A weight proportional to the path's length

Q3. The key mathematical step in deriving the path integral from the time-evolution operator is:

(a) Diagonalizing the Hamiltonian (b) Repeatedly inserting the completeness relation $\hat{I} = \int dx\,|x\rangle\langle x|$ (c) Expanding in a perturbation series (d) Applying the variational principle

Q4. The Trotter decomposition $e^{-i(\hat{A}+\hat{B})\epsilon} \approx e^{-i\hat{A}\epsilon}\,e^{-i\hat{B}\epsilon}$ has an error of order:

(a) $\epsilon$ (b) $\epsilon^2$ (c) $\epsilon^3$ (d) It is exact for all $\epsilon$

Q5. In the path integral, the dominant paths in the limit $S \gg \hbar$ are:

(a) The shortest paths between the endpoints (b) The paths near the classical trajectory (stationary action) (c) The paths with the largest kinetic energy (d) Random (Brownian-motion-like) paths with no preferred direction

Q6. The free-particle propagator in one dimension is:

(a) $K = \sqrt{m/(2\pi\hbar t)}\,\exp[-m(x_f - x_i)^2/(2\hbar t)]$ (b) $K = \sqrt{m/(2\pi i\hbar t)}\,\exp[im(x_f - x_i)^2/(2\hbar t)]$ (c) $K = \delta(x_f - x_i - pt/m)$ (d) $K = (1/L)\exp[ip(x_f - x_i)/\hbar]$

Q7. As $t \to 0^+$, the free-particle propagator approaches:

(a) Zero everywhere (b) A constant (c) $\delta(x_f - x_i)$ (d) A plane wave $e^{ikx}$

Q8. The phase of the free-particle propagator, $m(x_f - x_i)^2/(2\hbar t)$, is equal to:

(a) The kinetic energy divided by $\hbar$ (b) The classical action divided by $\hbar$ (c) The momentum times the distance, divided by $\hbar$ (d) The potential energy times the time, divided by $\hbar$

Q9. The condition $\delta S[x_{\text{cl}}] = 0$ (stationary action on the classical path) yields:

(a) The Schrödinger equation (b) The Heisenberg uncertainty principle (c) The Euler-Lagrange equations (Newton's second law) (d) The Born rule for measurement probabilities

Q10. For a macroscopic baseball ($S_{\text{cl}} \sim 10^{33}\hbar$), the path integral is dominated by:

(a) All paths equally — quantum interference is always important (b) Paths very close to the classical trajectory (c) Paths that are random walks (d) Two paths (the direct path and one reflection)

Q11. The path integral for the quantum harmonic oscillator can be evaluated exactly because:

(a) The oscillator has equally spaced energy levels (b) The action is quadratic in $x$ and $\dot{x}$, making the integral Gaussian (c) The wave functions are known analytically (Hermite polynomials) (d) The potential is bounded

Q12. At time $t = \pi/\omega$ (half the classical period), the QHO propagator becomes proportional to:

(a) $\delta(x_f - x_i)$ — identity (b) $\delta(x_f + x_i)$ — parity inversion (c) $\delta(x_f)$ — collapse to the origin (d) A constant — uniform distribution

Q13. The energy eigenvalues of the QHO can be extracted from the path integral propagator by:

(a) Finding the zeros of the propagator (b) Differentiating the propagator with respect to $\omega$ (c) Taking the trace (setting $x_f = x_i$, integrating) and identifying the spectral decomposition (d) Applying time-dependent perturbation theory

Q14. The Wick rotation $t \to -i\tau$ transforms the path integral phase $e^{iS/\hbar}$ into:

(a) $e^{-S_E/\hbar}$, a real suppression factor (b) $e^{S_E/\hbar}$, an exponentially growing factor (c) $e^{-iS_E/\hbar}$, still an oscillatory phase (d) $1$ (the phase is removed entirely)

Q15. The quantum partition function $Z = \text{Tr}(e^{-\beta\hat{H}})$ can be written as a path integral over:

(a) All paths from $x_i$ to $x_f$ in real time (b) All paths from $x_i$ to $x_f$ in imaginary time $\beta\hbar$ (c) All periodic paths $x(0) = x(\beta\hbar)$ in imaginary time (d) Only the classical path in imaginary time

Q16. The high-temperature limit of the QHO partition function $Z = 1/[2\sinh(\beta\hbar\omega/2)]$ gives:

(a) $Z \to e^{-\beta\hbar\omega/2}$ (zero-point energy only) (b) $Z \to k_BT/(\hbar\omega)$ (classical equipartition) (c) $Z \to 1$ (all states equally populated) (d) $Z \to \infty$ (divergent)

Q17. The Euclidean path integral for a free particle is mathematically equivalent to:

(a) Classical mechanics of a free particle (b) The diffusion equation / Brownian motion (c) The wave equation for sound (d) Maxwell's equations for electromagnetism

Q18. An instanton is:

(a) A quantum state with zero energy (b) A classical solution of the Euclidean equations of motion that mediates tunneling (c) A virtual particle that exists for an infinitesimal time (d) A perturbative correction to the ground-state energy

Q19. Feynman diagrams arise from the path integral through:

(a) Exact evaluation of the full non-Gaussian integral (b) The Wick rotation to imaginary time (c) Perturbative expansion of the non-Gaussian part and application of Wick's theorem (d) The WKB approximation to the propagator

Q20. Which of the following is NOT a system for which the path integral can be evaluated exactly (without approximation)?

(a) Free particle (b) Quantum harmonic oscillator (c) Particle in a uniform external force (d) Hydrogen atom (Coulomb potential)