Introduction to Quantum Mechanics · Problem 3.4

Formalism — David J. Griffiths

Physics — Quantum Mechanics · harmonic_oscillator_expectation

Problem

The most general wave function of a particle in the simple harmonic oscillator potential is

\Psi(x, t) = \sum_n c_n \psi_n(x) e^{-iE_n t/\hbar}.

Show that the expectation value of position is

\langle x \rangle = C \cos(\omega t - \phi),

where the real constants

C

and

\phi

are given by

Ce^{-i\phi} = \left(\sqrt{\frac{2\hbar}{m\omega}}\right) \sum_{n=0}^{\infty} \sqrt{n+1}\, c^*_{n+1} c_n.

Thus the expectation value of position for a particle in the harmonic oscillator oscillates at the classical frequency

\omega

(as you would expect from Ehrenfest's theorem; see problem 3.19(b)). *Hint:* Use Equation 3.114. As an example, find

C

and

\phi

for the wave function in Problem 2.40.