Topics#

Key Concepts#

• Stationary phase: oscillatory integrals dominated by \(\Phi'(x_0)=0\); in the path integral, this selects the classical path

• WKB wavefunctions: \(\psi \sim (1/\sqrt{p})\,\mathrm{e}^{\mathrm{i}\int p\,\mathrm{d}x/\hbar}\) with phase from Hamilton-Jacobi, amplitude from probability conservation

• Connection formulas: match oscillating and decaying solutions across turning points with \(\pi/4\) phase shifts

• Bohr-Sommerfeld rule: \(\oint p\,\mathrm{d}x = 2\pi\hbar(n + \tfrac{1}{2})\) quantizes bound-state energies

• Maslov index: the \(\tfrac{1}{2}\) from turning-point phases gives zero-point energy

Learning Objectives#

• Apply stationary phase to evaluate oscillatory integrals and explain why the classical path dominates the path integral

• Construct WKB wavefunctions in allowed and forbidden regions; use connection formulas at turning points

• Derive energy levels from the Bohr-Sommerfeld quantization rule and verify for the harmonic oscillator

• Explain tunneling rates from WKB and the correspondence principle as \(S/\hbar \to \infty\)

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).