Recall the harmonic oscillator ladder operators from Chapter 4: $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ and $\hat{a}^\dagger|n\rangle = \sqrt{n+1}|n+1\rangle$. Using Dirac notation, we can instantly compute matrix elements: $\langle m|\hat{a}|n\rangle = \sqrt{n}\,\delta_{m,n-1}$ and $\langle m|\hat