Annihilation and Creation operator problem

[Trogdor27]

I've looked through all my course notes, but I just don't even know where to start with this problem.

The problem:

$$\langle 1 \mid x^2 \mid 2 \rangle$$

Use the annihilation and creation form for x to obtain the above matrix element.

What I do know:

I know that $$x = \sqrt{\hbar / 2mw}(a^\dagger + a)$$But where do I go from here? Any help or pointers are much appreciated!

 

[Matterwave]

So...square that x...and then apply it to your states. What is the effect of the creation operator on the state 2? The effect of the annihilation operator? Lastly, consider that your basis states should be orthonormal.

 

[Trogdor27]

Ah - I didn't realize that 1 & 2 correspond to different states, I thought they were just numbers. It all makes sense now, and if I'm right the answer should be zero.

Thanks!