Principles of Quantum Mechanics · Problem 21.13

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-zeta-function

Problem

The functional determinant of the operator −∂t2+ω2-\partial_t^2 + \omega^2 on [0,T][0,T] with Dirichlet boundary conditions arises in the harmonic oscillator path integral. Use the zeta-function regularization or the eigenvalue product formula to show that det⁡(−∂t2+ω2)∝sinh⁡(ωT)ωT\det(-\partial_t^2 + \omega^2) \propto \frac{\sinh(\omega T)}{\omega T} and hence recover the harmonic oscillator propagator prefactor.

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