Hydrogen Atom in homogeneous magnetic vector potential

[VVS]

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

 

[mfb]

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.

 

[VVS]

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

 

[mfb]

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.

 

[VVS]

I am not satisfied with that answer. Go through the calculations before you make any judgement or show me a different proof.

 

[mfb]

I found a way to get the result without any calculations. Isn't that much more elegant?

 

[Vanadium 50]

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.