Principles of Quantum Mechanics · Problem 21.10

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-wkb

Problem

Show that in the semiclassical limit (ℏ→0\hbar \to 0), the path integral is dominated by the classical trajectory, and the leading correction gives the Van Vleck determinant. That is, derive the semiclassical approximation G(x,x′;t)≈12πiℏ∣∂2Scl∂x∂x′∣exp⁡(iScl(x,x′;t)ℏ−iπ2ν)G(x, x'; t) \approx \sqrt{\frac{1}{2\pi i\hbar} \left|\frac{\partial^2 S_{cl}}{\partial x \partial x'}\right|} \exp\left(\frac{i S_{cl}(x, x'; t)}{\hbar} - i\frac{\pi}{2}\nu\right) where SclS_{cl} is the classical action and ν\nu is the Maslov index.

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