Phy5645/HO problem1

Problem 1: Calculate the expectation value of x in eigenstate.

Solution:

We can compute the expectation value of x simply.

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

=\sqrt{\frac{\hbar}{2m\omega}}(\langle\Psi_n|\hat{a}\Psi_n\rangle+\langle\Psi_n|\hat{a}^{\dagger}\Psi_n\rangle)=\sqrt{\frac{\hbar}{2m\omega}}(\sqrt{n}\langle\Psi_n|\Psi_n-1\rangle+\sqrt{n+1}\langle\Psi_n|\Psi_n+1\rangle)=0</math>

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.

(Submitted by Team 4-Hang Chen)