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Homework Statement
The vector [itex]\psi =\psi_{n}[/itex] is a normalized eigenvector for the energy level [itex]E=E_{n}=(n+\frac{1}{2})\hbar\omega [/itex] of the harmonic oscillator with Hamiltonian [itex]H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}[/itex]. Show that:
[itex]E=\frac{\mathbb{E}_{\psi}[P^{2}]}{2m}+\frac{1}{2}m\omega^{2}\mathbb{E}_{\psi}[X^{2}][/itex]
Homework Equations
Time independent Schrodinger: [itex]H\psi=E\psi[/itex]
The Attempt at a Solution
Am I wrong in thinking it's as simple as taking the expectation of both sides? I feel like I must be as that gives [itex]\mathbb{E}_{\psi}[E]=\frac{\mathbb{E}_{\psi}[P^{2}]}{2m}+\frac{1}{2}m\omega^{2}\mathbb{E}_{\psi}[X^{2}][/itex] which isn't quite right.
But I can't see how else I'd do it?
Thanks for your help in advance!
EDIT: I believe the extra information is needed to answer the rest of the question (this is just part 1).