Introduction to Quantum Mechanics · Problem 3.39

Formalism — David J. Griffiths

Physics — Quantum Mechanics · harmonic_oscillator_matrix

Problem

Find the matrix elements

\langle n|x|n'\rangle

and

\langle n|p|n'\rangle

in the (orthonormal) basis of stationary states for the harmonic oscillator (Equation 2.68). You already calculated the "diagonal" elements

(n = n')

in Problem 2.12; use the same technique for the general case. Construct the corresponding (infinite) matrices,

\mathsf{X}

and

\mathsf{P}

. Show that

(1/2m)\mathsf{P}^2 + (m\omega^2/2)\mathsf{X}^2 = \mathsf{H}

is diagonal, in this basis. Are its diagonal elements what you would expect? *Partial answer:*

\langle n | x | n' \rangle = \sqrt{\frac{\hbar}{2m\omega}}\left(\sqrt{n'}\delta_{n,n'-1} + \sqrt{n}\delta_{n',n-1}\right). \tag{3.114}