Phy5645/HO problem1

Calculate the expectation value of x in eigenstate.

Solution:

We can compute the expectation value of x simply.

$\langle \Psi _{n}|x|\Psi _{n}\rangle ={\sqrt {\frac {\hbar }{2m\omega }}}\langle \Psi _{n}|hat{a}+hat{a}^{\dagger }|\Psi _{n}\rangle$

We should have seen that coming. Since each term in the operator changes the eigenstate, the dot product with the original (orthogonal) state must give zero