For the harmonic oscillator,

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Quantum Mechanics · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The path integral

Problem

For the harmonic oscillator, L=12mx˙2−12mω2x2L=\tfrac12m\dot x^2-\tfrac12m\omega^2x^2, take 0<ωt<π0<\omega t<\pi. (a) Find the classical path xcl(t′)x_{\rm cl}(t') that runs from x′x' at t′=0t'=0 to xx at t′=tt'=t. (b) Show that its action is Scl=mω2sin⁡ωt[(x2+x′2)cos⁡ωt−2xx′]S_{\rm cl}=\dfrac{m\omega}{2\sin\omega t}\left[(x^2+x'^2)\cos\omega t-2xx'\right]. (c) Show that SclS_{\rm cl} becomes the free action m(x−x′)2/2tm(x-x')^2/2t as ω→0\omega\to0. Then write the full oscillator propagator and say where its dependence on the endpoints sits, and why. (d) Explain what goes wrong with part (a) at ωt=π\omega t=\pi.

Related theory

Browse the problem library