3.3 Stationary Phase

3.3 Stationary Phase#

Overview#

The path integral sums over all paths weighted by \(\mathrm{e}^{\mathrm{i}S/\hbar}\), yet macroscopic objects follow definite classical trajectories. The stationary-phase principle resolves this: for \(S \gg \hbar\), all but a narrow tube of paths around the classical trajectory cancel by destructive interference. This unit makes the resolution quantitative — first as the mathematical SPA in §3.3.1, then as the WKB wavefunction it implies in §3.3.2, then as the Bohr-Sommerfeld quantization rule for bound states in §3.3.3 — and shows that bound-state energies arise from constructive interference of classical periodic orbits.

Topics#

Lesson

Title

Core Question

3.3.1

Stationary Phase Approximation

How does classical mechanics emerge from the path integral as \(\hbar\to 0\)?

3.3.2

WKB Approximation

What approximate wavefunction does the WKB ansatz build, and how does it predict tunneling?

3.3.3

Bohr-Sommerfeld Quantization

Why do bound-state energies arise from constructive interference of periodic orbits?

Key Concepts#

  • Stationary-phase approximation: oscillatory integrals \(\int\mathrm{e}^{\mathrm{i}\Phi/\hbar}\mathrm{d}x\) are dominated by points \(\Phi'(x_{0}) = 0\); in the path integral this picks out the classical path.

  • WKB wavefunction: \(\psi \sim p(x)^{-1/2}\,\mathrm{e}^{\pm\mathrm{i}\int p\,\mathrm{d}x/\hbar}\) in classically allowed regions; exponentially decaying in forbidden regions.

  • Connection formulas: match oscillating and decaying WKB solutions across turning points with a \(\pi/4\) phase shift (Maslov correction).

  • WKB tunneling: \(T \approx \mathrm{e}^{-2\gamma}\) with \(\gamma = \int\kappa\,\mathrm{d}x/\hbar\).

  • Bohr-Sommerfeld rule: \(\oint p\,\mathrm{d}x = 2\pi\hbar(n + 1/2)\) for two soft turning points; gives the exact harmonic-oscillator and hydrogen spectra.

  • Periodic-orbit interpretation: discrete energies are the energies at which a classical periodic orbit interferes constructively with itself under repeated traversal.

Learning Objectives#

  • Apply stationary phase to evaluate oscillatory integrals and explain why the classical path dominates the path integral as \(\hbar\to 0\).

  • Construct WKB wavefunctions in allowed and forbidden regions; use connection formulas at soft turning points to compute tunneling probabilities.

  • Derive the Bohr-Sommerfeld rule from single-valuedness and verify it for the harmonic oscillator and the particle in a box.

  • Articulate the path-integral origin of Bohr-Sommerfeld as constructive interference of periodic orbits, and the correspondence principle as \(S/\hbar \to \infty\).

Project#

Project: Semiclassical Theory of Quantum Chaos: Gutzwiller Trace Formula#

Objective: Implement the Gutzwiller trace formula, which connects quantum energy levels to classical periodic orbits, and test it on a simple chaotic billiard system to uncover the hidden classical structure in quantum chaos.

Background: One of the deepest insights of semiclassical physics is that quantum spectra encode information about classical dynamics. The Gutzwiller trace formula expresses the quantum density of states as a sum over all classical periodic orbits, weighted by the actions and stability properties (monodromy matrices) of those orbits. This formula is a frontier tool for understanding quantum chaos: systems whose classical counterparts are chaotic (exponential sensitivity to initial conditions) still possess a dense, deterministic quantum spectrum, yet that spectrum bears the mathematical fingerprint of the underlying chaotic dynamics. Applications range from quantum dots and microwave cavities to nuclear physics.

Suggested Approach:

  • Start with a simple integrable system (2D harmonic oscillator): find its classical periodic orbits analytically and compute the Gutzwiller formula. Verify that it reproduces the known quantum spectrum.

  • Study a 2D billiard system (e.g., square or circular billiard): compute its classical periodic orbits numerically by ray tracing (bounce maps).

  • For each periodic orbit, calculate the action \(S_p = \oint p \, \mathrm{d}q\) and the stability eigenvalues (Lyapunov exponents from the monodromy matrix).

  • Construct the Gutzwiller trace: \(\rho(E) = \rho_0(E) + \sum_p \rho_p(E)\), where \(\rho_0\) is the smooth part and \(\rho_p\) is the oscillating contribution from periodic orbit \(p\).

  • Compare to the quantum spectrum obtained by diagonalizing the Hamiltonian numerically in a finite domain. Do short periodic orbits dominate? Which orbits contribute most to oscillations?

  • Explore a chaotic billiard (e.g., stadium or Sinai billiard): show that the classical periodic orbits are now exponentially proliferating, yet the Gutzwiller formula still provides a quantitative account of spectral oscillations.

Expected Deliverable: A research report (5–10 pages) with code. Include: (i) classical billiard dynamics and periodic orbit enumeration, (ii) derivation and implementation of the Gutzwiller formula, (iii) quantum spectrum (from numerics) vs. semiclassical prediction (from Gutzwiller), (iv) quantitative comparison: residue analysis, quality of fit, (v) physical interpretation: why do classical periodic orbits predict quantum levels?, (vi) discussion of quantum chaos and the classical-quantum correspondence.

Key References: M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, 1990); D. Ullmo, Rep. Prog. Phys. 71, 026001 (2008) [quantum chaos and periodic orbits]; E. B. Bogomolny et al., Phys. Rev. Lett. 85, 2486 (2000).