Hydrogen Atom Matrix Elements Related to Transition Probability

In summary, the problem involves finding the matrix element <U210|z|U100> in terms of the hydrogen atom's quantum numbers. The method involves using z = rcosθ and converting to spherical coordinates to evaluate a volume integral, without needing to integrate dz. This method is also applicable for finding transition probabilities under perturbation theory.
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Homework Statement


Evaluate the matrix element <U210|z|U100> where by |Unlm> 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!
 
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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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1. What are hydrogen atom matrix elements related to transition probability?

The hydrogen atom matrix elements related to transition probability are mathematical quantities that describe the likelihood of an electron transitioning from one energy level to another within a hydrogen atom. These elements are important in understanding the behavior and properties of hydrogen atoms.

2. How are these matrix elements calculated?

The matrix elements are calculated using quantum mechanical principles and equations. These calculations take into account the energy levels and wavefunctions of the electron in the hydrogen atom.

3. What is the significance of transition probability in hydrogen atoms?

The transition probability in hydrogen atoms is important for understanding the absorption and emission of light by these atoms. It also helps to explain the spectral lines observed in hydrogen emission spectra, which are used in various fields such as astronomy and chemistry.

4. How does the transition probability change with different energy levels?

The transition probability generally increases with increasing energy levels. This is because higher energy levels have a larger transition probability due to the increased distance between the electron and the nucleus, allowing for a greater chance of transition.

5. Are these matrix elements only applicable to hydrogen atoms?

No, while they were first derived for hydrogen atoms, these matrix elements can also be applied to other atoms and molecules. However, the calculations may be more complex for multi-electron systems.

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