Phy5645/HO problem2

Problem 2:

The expectation value of p in eigenstate.

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

$==-i{\sqrt {\frac {m\hbar \omega }{2}}}(\langle \Psi _{n}|{\hat {a}}\Psi _{n}\rangle -\langle \Psi _{n}|{\hat {a}}^{\dagger }\Psi _{n}\rangle )$

$==-i{\sqrt {\frac {m\hbar \omega }{2}}}({\sqrt {n}}\langle \Psi _{n}|\Psi _{n}-1\rangle -{\sqrt {n+1}}\langle \Psi _{n}|\Psi _{n}+1\rangle )$

$=0$