# Homework Help: Expectation of Position of a Harmonic Oscillator

1. Jan 17, 2013

### Sekonda

Hey,

My question is on determing the expectation value of position of the Harmonic Oscillator using raising and lowering operators, the question is part d) below:

I have determined the position operator to be:

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

and so the expectation value becomes:

$$\sqrt{\frac{\hbar}{2m\omega}}<E_{1}|(a+a^{\dagger})|E_{0}>$$

Though this is the bit I'm not sure about, I think only the a-dagger operator acts to increase the E-zero ket and the 'a' operator just causes the state to go to zero thus:

$$\sqrt{\frac{\hbar}{2m\omega}}=<E_{1}|\hat{x}|E_{0}>$$

Can someone confirm this?

Cheers,
SK

2. Jan 17, 2013

### vela

Staff Emeritus
Looks good.