# Map energy eigenstates to cartesian unit vectors - Harmonic Osillator

1. Apr 9, 2012

### PhysicsGente

1. The problem statement, all variables and given/known data
Evaluate the matrix elements
$$x_{nn'} = \left<n\left|x\right|n'\right>$$
and
$$p_{nn'} = \left<n\left|p\right|n'\right>$$
and map the energy eigenstates
$$\left|n\right>$$
to Cartesian unit vectors.

2. Relevant equations

$$x = \sqrt{\frac{\hbar}{2m \omega}}\left(a+a^{\dagger}\right)$$
$$p = -i \sqrt{\frac{\hbar m\omega}{2}}\left(a-a^{\dagger}\right)$$

3. The attempt at a solution

I have

$$x_{nn'} = \sqrt{\frac{\hbar}{2m \omega}}\left(\sqrt{n'}\left<n|n'-1\right>+\sqrt{n'+1}\left<n|n'+1\right>\right)$$

and

$$p_{nn'} = -i \sqrt{\frac{\hbar m\omega}{2}}\left(\sqrt{n'}\left<n|n'-1\right>-\sqrt{n'+1}\left<n|n'+1\right>\right)$$

But I'm confused with the second part of the question. For example, I believe that mapping a state vector into position space would mean to get the projection of the state vector in position space meaning that one has to take the inner product <x|ψ> = ψ(x). But I don't see how i can do this with Cartesian unit vectors.