Principles of Quantum Mechanics · Problem 7.3.4

The Harmonic Oscillator — R. Shankar

Physics — Quantum Mechanics · harmonic-oscillator-matrix-elements-xp

Problem

Using Eqs. (7.3.23)--(7.3.25), show that ⟨n′∣X∣n⟩=(ℏ2mω)1/2[δn′,n+1(n+1)1/2+δn′,n−1n1/2]\langle n'|X|n\rangle = \left(\frac{\hbar}{2m\omega}\right)^{1/2} \left[\delta_{n',n+1}(n+1)^{1/2} + \delta_{n',n-1} n^{1/2}\right] ⟨n′∣P∣n⟩=(mωℏ2)1/2i[δn′,n+1(n+1)1/2−δn′,n−1n1/2]\langle n'|P|n\rangle = \left(\frac{m\omega\hbar}{2}\right)^{1/2} i\left[\delta_{n',n+1}(n+1)^{1/2} - \delta_{n',n-1} n^{1/2}\right]

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