Hydrogen Atom Matrix Elements Related to Transition Probability

[The Head]

Homework Statement

Evaluate the matrix element <U₂₁₀|z|U₁₀₀> where by |U_nlm> we mean the hydrogen atom orbital with it's quantum numbers.

Homework Equations

The Attempt at a Solution

So where I'm getting stuck is on the integral, because the "U" portion of the wave function is given in terms of r and theta, whereas we are putting the z operator between these two U functions.

So we get ∫U₂₁₀(z)U₁₀₀dz

This is where I get stuck. I tried converting z to spherical coordinates, using z=r*cosθ, but then dz=cosθ dr - r*sinθdθ. Thus, when I integrate the radial portion, I still have θ unevaluated (still a variable) and vice versa. Then I tried converting r and θ to z and just integrating over dz. But when I put this integrand into Wolfram's online integral calculator, it seems too difficult to evaluate (and I wouldn't have any clue by hand).

I'm wondering, is this even the correct method in the first place? It is just confusing to me to evaluate a 3-D hydrogen atom only along z. Usually the text I uses doesn't give impossible integrals, so I suspect I am setting it up incorrectly.

To give some context, later in the problem, we are going to evaluate transition probabilities under perturbation theory, which also employs a z operator.

Thank you!

 

[TSny]

Yes, use z = rcosθ and express everything in spherical coordinates. You will not need to use an expression for dz. You are integrating a volume integral, so make sure you express your volume element dV in spherical coordinates.

See here for example.