Principles of Quantum Mechanics · Problem 7.3.5

The Harmonic Oscillator — R. Shankar

Physics — Quantum Mechanics · harmonic-oscillator-expectation-values-uncertainty

Problem

^*

Using the symmetry arguments from Exercise 7.3.3 show that

\langle n|X|n\rangle = \langle n|P|n\rangle = 0

and thus that

\langle X^2\rangle = (\Delta X)^2

and

\langle P^2\rangle = (\Delta P)^2

in these states. Show that

\langle 1|X^2|1\rangle = 3\hbar/2m\omega

and

\langle 1|P^2|1\rangle = \frac{3}{2}m\omega\hbar

. Show that

\psi_0(x)

saturates the uncertainty bound

\Delta X \cdot \Delta P \geq \hbar/2