Map energy eigenstates to cartesian unit vectors - Harmonic Osillator

[PhysicsGente]

Homework Statement

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.

Homework Equations

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

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.

 

[mark726]

To map the energy eigenstates to Cartesian unit vectors, we can use the following equations:

\left|n\right> = \frac{1}{\sqrt{n!}}\left(a^{\dagger}\right)^n\left|0\right>

where \left|0\right> is the ground state. By substituting this into the expressions for x and p, we can see that the energy eigenstates \left|n\right> correspond to the Cartesian unit vectors \hat{x} and \hat{p}, respectively. This means that the energy eigenstate \left|n\right> is mapped to the unit vector \hat{x} in the x direction, and the energy eigenstate \left|n'\right> is mapped to the unit vector \hat{p} in the p direction. So, for example, the state \left|0\right> is mapped to the unit vector \hat{x}, \left|1\right> is mapped to the unit vector \hat{p}, and so on.