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Hydrogen Atom Matrix Elements Related to Transition Probability

  1. Nov 27, 2013 #1
    1. The problem statement, all variables and given/known data
    Evaluate the matrix element <U210|z|U100> where by |Unlm> we mean the hydrogen atom orbital with it's quantum numbers.


    2. Relevant equations



    3. 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!
     
  2. jcsd
  3. Nov 27, 2013 #2

    TSny

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    Homework Helper
    Gold Member

    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.
     
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