Virial Theorem and Simple Harmonic Oscillator

[njdevils45]

Homework Statement

Show that the virial theorem holds for all harmonic-oscillator states. The identity given in problem 5-10 is helpful.

Homework Equations

Identity given: ∫ξ²H²_n(ξ)e^-ξ₂dξ = 2ⁿn!(n+1/2)√pi

P.S the ξ in the exponent should be raised to the 2nd power. So it should look like ξ² but for whatever reason it's just not coming out like that.

The Attempt at a Solution

I'm guessing I find the value of E for a given value n, which I'm pretty sure is E_n = (n+½)ħω and I get that the expected value of the potential should equal that, I just don't know how to set up the problem to allow me to solve for <V>

 

[Orodruin]

How do you express the expectation value of any observable?