Tricky Integral, H-Atom Dipole Transition Matrix Elements

In summary, the conversation discusses a problem involving the evaluation of an integral for computing dipole moment matrix elements. The integral is in spherical coordinates and requires integration by parts. The participants also discuss a related integral and clarify the convention for integrating over angles in spherical coordinates. The conversation ends with the problem being solved.
  • #1
logic smogic
56
0
1. Problem

Evaluate

[tex]\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} z^{2} e^{-A \sqrt{x^{2}+y^{2}+z^{2}}} dxdydz[/tex]

2. Useful Formulae

none

3. Attempt at Solution

Well, this is part of a much larger problem. I am trying to compute the dipole moment matrix elements for a Hydrogen 2P -> 1S transition, and it involves this integral. In particular, this is the "z-component" of the |2,1,0> -> |1,0,0> transition - I've already found the x- and y-components to be zero for this transition.

I've spent nearly an hour looking through online integral tables, and I've found nothing useful. Can anyone help? Thanks!
 
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  • #2
switch to spherical coordinates and it should be doable.
It will probably take an integration by parts once or twice. (don't forget your jacobian)
 
  • #3
I started in spherical, and had a tough time with it. But maybe I was turning to the definite integral tables too early. Looks like if I do integration by parts two (or maybe three?) times, it should take a familiar form.

Alright, I'll give it a shot.
 
  • #4
Great, I got it. Thanks.

Quick question, though. What is:

[tex]\int_{0}^{2 \pi} \sin \phi d \phi[/tex]

Isn't it zero? But then all of my matrix elements would go to zero when evaluated in spherical coordinates (where [itex]\phi[/itex] is evaluate from [itex]0 \rightarrow 2 \pi[/itex]). It seems like cheating to evaluate it half-way and multiply by 2, giving 4 - but is that the trick?
 
  • #5
logic smogic said:
Great, I got it. Thanks.

Quick question, though. What is:

[tex]\int_{0}^{2 \pi} \sin \phi d \phi[/tex]

Isn't it zero? But then all of my matrix elements would go to zero when evaluated in spherical coordinates (where [itex]\phi[/itex] is evaluate from [itex]0 \rightarrow 2 \pi[/itex]). It seems like cheating to evaluate it half-way and multiply by 2, giving 4 - but is that the trick?

that integral over 2pi is zero; but you integrate theta over 2 pi, and phi over pi. Does that make sense?
phi is the angle coming from the z axis, and theta is the angle in the xy plane. only one of those 2 angles needs to go from 0 to 2pi, and the other from 0 to pi; by convention this is phi and not theta --> and that is tied to the jacobian r^2 * sinphi dr dphi dtheta
 
  • #6
Ah, I must've had my convention backwards. That fixes everything - thanks!
 
  • #7
No prob!
 

1. What is a tricky integral?

A tricky integral is a mathematical integral that is difficult to solve or requires advanced techniques to solve. It often involves complex functions and may not have a closed-form solution.

2. What are H-Atom Dipole Transition Matrix Elements?

H-Atom Dipole Transition Matrix Elements are mathematical quantities used in the study of atomic and molecular physics. They represent the strength of the interaction between an electron and a photon during a transition from one energy state to another in a hydrogen atom.

3. Why are H-Atom Dipole Transition Matrix Elements important?

H-Atom Dipole Transition Matrix Elements are important because they provide valuable information about the properties and behavior of atoms and molecules. They are used in various applications, such as spectroscopy, laser technology, and quantum mechanics.

4. What makes calculating H-Atom Dipole Transition Matrix Elements tricky?

Calculating H-Atom Dipole Transition Matrix Elements can be tricky because it involves solving complex integrals that may not have a closed-form solution. Additionally, the integrands may involve complicated functions and require advanced mathematical techniques to solve.

5. How are H-Atom Dipole Transition Matrix Elements calculated?

H-Atom Dipole Transition Matrix Elements are typically calculated using a combination of analytical and numerical techniques. This involves breaking down the integral into smaller, more manageable parts and using mathematical approximations to solve them. Advanced computational methods are also used for more complex integrands.

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