Hydrogen Atom in homogeneous magnetic vector potential

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Discussion Overview

The discussion revolves around the quantum mechanical analysis of a Hydrogen Atom in a homogeneous magnetic vector potential. Participants explore the implications of this scenario, particularly focusing on energy eigenvalues and wave function behavior under perturbation theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents an analysis of a Hydrogen Atom in a homogeneous magnetic vector potential, expressing concerns about the feasibility of such a field.
  • Another participant expresses skepticism about observing deviations from the regular hydrogen atom due to gauge invariance.
  • A participant reports obtaining imaginary correction terms for energy eigenvalues using perturbation theory, indicating a rapid decay of the wave function, which raises concerns about the validity of the results.
  • Another participant suggests that the wave function may experience phase changes but asserts that energy eigenvalues should remain unchanged, questioning the initial findings.
  • A participant challenges the assertion of no deviation, urging for a thorough review of the calculations and alternative proofs.
  • One participant claims to have derived results without calculations, suggesting a more elegant approach.
  • Another participant notes that if the magnetic vector potential is constant and the magnetic field is zero, the energy levels would remain the same as those of a hydrogen atom in vacuum.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the analysis, particularly regarding the behavior of energy eigenvalues and wave functions. There is no consensus on the validity of the results or the interpretation of the findings.

Contextual Notes

Participants highlight potential issues with the perturbation theory results, including the possibility of incorrect calculations and the implications of gauge invariance. The discussion remains open to interpretation and further exploration.

Who May Find This Useful

This discussion may be of interest to those studying quantum mechanics, particularly in the context of magnetic fields and perturbation theory, as well as researchers exploring theoretical models of atomic systems.

VVS
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Hey!

I did an quantum mechanical analysis of a Hydrogen Atom in a homogeneous magnetic vector potential (I know that it might be impossible to create this kind of field) out of curiousity. I showed it to some professors of mine, but they all said that they don't have time. So I decided to post it online here for checkup.

Here is the analysis View attachment Hydrogen_in_Vector_Potentialv2.pdf

And here is a short appendix about the evaluation of the derivatives of Spherical Harmonics.
https://www.physicsforums.com/attachment.php?attachmentid=67418&d=1394285330

I would really appreciate it if you guys could have a look at the analysis and point out mistakes if there are any.

thanking you in advance
VVS
 
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What is the result?

I would be surprised to see any deviation from the regular hydrogen atom, as homogeneous vector potentials should be covered by gauge invariance.
 
Hey mfb!
I get a very weird result. Using perturbation theory you get imaginary corrections terms for the energy eigenvalues. That is as far as I know possible. But it basically means that the wave function decays to zero. But I am more concerned with the rate at which it decays, which according to the equations is VERY VERY FAST.
Something must be wrong.
VVS
 
But it basically means that the wave function decays to zero.
Or your perturbation theory does not give proper results.

Something must be wrong.
Indeed, as the result is clear: there is no deviation. The wavefunction might get phase changes, but the energy eigenvalues stay the same.
 
I am not satisfied with that answer. Go through the calculations before you make any judgement or show me a different proof.
 
I found a way to get the result without any calculations. Isn't that much more elegant?
 
Here's a second calculation: if A is constant, B = 0. If B is zero, there's no Zeeman effect, so you get the same energy levels as a hydrogen atom in vacuum.
 
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