Srednicki — Quantum Field Theory · Problem 7.4

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · path-integral-harmonic-oscillator

Problem

Consider a harmonic oscillator in its ground state at

t = -\infty

. It is then subjected to an external force

f(t)

. Compute the probability

|\langle 0|0\rangle_f|^2

that the oscillator is still in its ground state at

t = +\infty

. Write your answer as a manifestly real expression, and in terms of the Fourier transform

\widetilde{f}(E) = \int_{-\infty}^{+\infty} dt\, e^{iEt}f(t)

. Your answer should not involve any other unevaluated integrals.