Principles of Quantum Mechanics · Problem 8.6.3

The Heisenberg Uncertainty Relations — R. Shankar

Physics — Quantum Mechanics · harmonic-oscillator-eigenfunctions-propagator

Problem

We know that given the eigenfunctions and the eigenvalues we can construct the propagator:

U(x, t; x', t') = \sum_n \psi_n(x) \psi_n^*(x')\, e^{-iE_n(t-t')/\hbar} \tag{8.6.15}

Consider the reverse process (since the path integral approach gives

U

directly), for the case of the oscillator. (1) Set

x = x' = t' = 0

. Assume that

A(t) = (m\omega/2\pi i\hbar \sin\omega t)^{1/2}

for the oscillator. By expanding both sides of Eq.\ (8.6.15), you should find that

E = \hbar\omega/2,\, 5\hbar\omega/2,\, 9\hbar\omega/2, \ldots

, etc. What happened to the levels in between? (2) (Optional). Now consider the extraction of the eigenfunctions. Let

x = x'

and

t' = 0

. Find

E_0

E_1

|\psi_0(x)|^2

, and

|\psi_1(x)|^2

by expanding in powers of

\alpha = \exp(i\omega t)