Kinetic and potential energy of harmonic oscillator

[masudr]

Homework Statement

Use ladder operator methods to determine the kinetic and potential energy of eigenstates of the harmonic oscillator.

Homework Equations

$$H=\frac{p^2}{2m} + \frac{1}{2}m\omega x^2$$
$$x=\sqrt{\frac{\hbar}{2m \omega}}(a+a^{\dagger})$$

The Attempt at a Solution

So I squared x, and then substituted it in the expression for V:

$$\langle n | V | n \rangle$$

I ended up getting $V=\frac{1}{2}(n+\frac{1}{2})\hbar \omega$. This is one half of the energy expectation, so KE must be the same.

I just don't have any references with me to confirm if this is right. Can anyone tell me? Thanks in advance. I could write out what I got V as, if someone wants, but if the answer is right, there may not be much point.

 

[kuruman]

It is correct. You can verify your calculations independently using the virial theorem.