Undergrad Finding matrices of perturbation using creation/annihilation operators

[Keru]

"Given a 3D Harmonic Oscillator under the effects of a field W, determine the matrix for W in the base given by the first excited level"

So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result:

W = 2a_z² - a_x² - a_y² + 2a_z^{+ 2} -a_x^{+ 2} -a_y^{+ 2} + 2{a_z, a_z⁺}) - {a_x, a_x⁺}) - {a_y, a_y⁺}

As for the following part, I've proceeded like this, which I'm pretty sure it's wrong at some point.

In terms of the ladder operator, I'm using the basis:
|n_x, n_y,n_z>

With n_i corresponding to the level of excitement on a certain coordinate. I have taken the first excited level to be:

|1,0,0>

I generally find the matrix elements like:

M_i,j = <i|M^|j>

So my thought has been, I have to find them in the basis |1,0,0>, let's just:

<1,0,0|W|1,0,0>

Which looks terrible because that's clearly a 1x1 "matrix".
What am I missing? Do the matrix actually correspond to |1,0,0>, |0,1,0>, |0,0,1> since all these possible combinations correspond to the first energy state? Is that simply not the basis I should be using?

Any help would be greatly appreciated.

 

[hilbert2]

Where is this from and what's the operator $\hat{W}(\hat{x},\hat{y},\hat{z})$ in terms of the position operators $\hat{x},\hat{y},\hat{z}$ ?

A single energy eigenstate (like the first excited level) obviously can't be a basis for the system, you need an infinite number of states.

 

[Keru]

Where is this from and what's the operator $\hat{W}(\hat{x},\hat{y},\hat{z})$ in terms of the position operators $\hat{x},\hat{y},\hat{z}$ ?

A single energy eigenstate (like the first excited level) obviously can't be a basis for the system, you need an infinite number of states.

In terms of position operators it is:
W = 2z² - x² - y²
and it comes from the electric cuadrupole.

So, there's something I'm not getting right then. I'll try to translate the exercise as precisely as possible:
"Determine the matrix of W in the base given by the states of the first energy excitation". Should I do all in terms of position operators and forget about creation/annihilation operators? The previous section asked us to express it in terms of them, but this one does not explicitly tell us to use them, so maybe it's just nonsensical trying to do it like i was?
Sorry if the question seems obvious I am pretty much new at this.

 

[hilbert2]

If the $\hat{W} = 2\hat{z}^2 - \hat{x}^2 - \hat{y}^2$ is supposed to be a potential energy (you first have to multiply it with something of dimensions [energy]/[length]^2 to make it have dimensions of energy), then it doesn't have a ground state at all because the potential does not have a lower limit (it can be as negative as you want if you increase $x$ or $y$ enough). Let's see what others say about this.