Introduction to Quantum Mechanics · Problem 2.49

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · harmonic_oscillator_exact

Problem

(a) Show that

\Psi(x, t) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp\left[-\frac{m\omega}{2\hbar}\left(x^2 + \frac{x_0^2}{2}\left(1 + e^{-2i\omega t}\right) + \frac{i\hbar t}{m} - 2x_0 x e^{-i\omega t}\right)\right]

satisfies the time-dependent Schrödinger equation for the harmonic oscillator potential (Equation 2.44). Here

x_0

is any real constant with the dimensions of length. (b) Find

|\Psi(x, t)|^2

, and describe the motion of the wave packet. (c) Compute

\langle x \rangle

and

\langle p \rangle

, and check that Ehrenfest's theorem (Equation 1.38) is satisfied.