What happens if you try to apply Maxwell's Equations to this quantum mechanical system?

Question

In another post, we discussed the oscillating charge in a hydrogen atom and the weight of opinion seemed to be that there is indeed an oscillating charge when you consider the superposition of the 1s and 2p states. One of the correspondents (freecharly) went a little farther and said that Schroedinger believed this oscillating charge to be the source of radiation. I wonder if the actual calculation bears this out? Specifically, in the case of the hydrogen atom in this particular superposition, do you get the correct decay times for the superposition of states if you apply Maxwell's equations to the oscillating charge and assume that as the system loses energy by radiation, the "probability" flows from the 2p to the 1s state in accordance with the amount of energy remaining in the system?

EDIT: Some people are objecting in different ways to the basic premise of the question, so let me make it a little more specific: I am not asking if hydrogen atoms ACTUALLY EXIST in a particular superposition of these states. (I may ask that in another question.) What I am asking here is IF you take (just to be specific) a 50-50 superposition of the 1s and 2p states, and apply Maxwell's equations to the oscillating charge, AND you assume that as the atom radiates the probability drains from the excited state to the ground state in such a way as to maintain conservation of energy...IF you do all those things, do you get a result that is consistent with standard QM?

Answer 1

I suspect the results would be correct (at least, approximately), as Barut developed his "self-field electrodynamics" (see, e.g. http://phys.lsu.edu/~jdowling/publications/Barut89b.pdf) and claimed results very close to those of QED. In self-field electrodynamics, radiation is created by charge density related to the wave function in a standard way (for the Dirac field).

Answer 2

I am disappointed that no one in this discussion group has been able to post a definitive answer as to whether the semi-classical calculation, applying Maxwell's equations to the quantum-mechanically oscillating charge, gives the correct result for the emission of radiation from an excited hydrogen atom. I appreciate akhmeteli's reference to a related publication but it does not directly address this question. So I am going to have to answer this question myself to the best of my ability by demonstrating a "back-of-the-envelope" type calculation.

I said I wanted to consider the 50-50 superposition of the 1s and 2p states. So first we need to know the maximum dipole moment of the superposition. I found the result on this University of Texas website by Prof. Richard Fitpatrick. I think I am interpreting it correctly when I say that the maximum charge displacement is 0.4 angstroms (about 75% of the standard radius of the ground state).

Then we need the frequency of the oscillation. Of course, this is the difference frequency corresponding to the 10.5 eV energy difference of the states, or 1.6 x 10^16 rad/sec.

Now we can calculate the acceleration. The easiest way to do this is to pretend it is uniform circular motion and use w^2*r. I get an acceleration of 10^22/m-sec^2. (Since it is actually harmonic motion and not circular, this will give us an error factor of 2 in the final result.)

Now I just plug this acceleration into the Larmour formula. You can find the Larmour formula anywhere on the internet, but I have simply converted all the physical constants into numerical values, and it comes to

   **Radiated Power  =   6 x 10^-54 a^2**

You can see that when I plug my value for acceleration into this formula I get a total radiate power of 6 x 10^-10 watts. This we divide by 2 to account for the harmonic motion vs circular.

Is this the correct power? We have to convert to "transition time" to find out. The total energy of the excited state is 10.5 (call it 10) eV which comes to 1.6 x 10^-18 Joules. Dividing the energy by the power, we get the lifetime of the excited state as just about 5 nanoseconds. Or maybe I'm wrongt about the energy and I should be taking it as half (because of the superposition) which would then give me a lifetime of 2.5 nanoseconds. Something like that.

This may not be exact but I think it's pretty much in the ballpark.

Answer 3

Good question and very interesting topic that I really appreciate.

I wonder if the actual calculation bears this out?

The superposition of 1s and 2p states, or even (n,l) and (m,l+/-1) states (selection rules) may generate indeed an oscillating charge in principle. Like for instance 1s and 2p stationary states provide an oscillating charge:

$$ \Psi_{1s}^*(\boldsymbol{r},t) \Psi_{2p}(\boldsymbol{r},t) \propto e^{-i (E_{1s} - E_{2p}) t /\hbar} \Phi_{1s}^*(\boldsymbol{r}) \Phi_{2p}(\boldsymbol{r}) $$

Alone it is useless to compute the electromagnetic field due to the transition of states (since it does not lead to a magnetic vector potential, required for the electromagnetic field).

One of the correspondents (freecharly) went a little farther and said that Schroedinger believed this oscillating charge to be the source of radiation

It is, in a formalism. Taking Schiff book on quantum mechanics (1957) and Roger Boudet's theory ("Relativistic Transitions in the Hydrogenic Atoms") you have a transition current:

$$ \boldsymbol{j} \propto \Psi_{1s}^* \boldsymbol{\nabla} \Psi_{2p} - \Psi_{2p} \boldsymbol{\nabla} \Psi_{1s}^* \propto \left( \Phi_{1s}^* \boldsymbol{\nabla} \Phi_{2p} - \Phi_{2p} \boldsymbol{\nabla} \Phi_{1s}^* \right)e^{-i (E_{1s} - E_{2p}) t /\hbar} $$

With this you can compute a magnetic vector potential and the electric field of the electromagnetic wave corresponding to, in this case, the Lyman transition (simple computations here: https://hal.science/hal-04369544/document, preprint draft only or hard one here: enter link description here). You can also derive Einstein's spontaneous coefficient with this. This is a powerful tool but discarded quickly by Pauli neglecting the "electron cloud" aspect of Schrodinger theory (Solvay congress).

Specifically, in the case of the hydrogen atom in this particular superposition, do you get the correct decay times for the superposition of states if you apply Maxwell's equations to the oscillating charge and assume that as the system loses energy by radiation, the "probability" flows from the 2p to the 1s state in accordance with the amount of energy remaining in the system?

I have no idea how to compute the decay rate, in this theory that I showed you, you can only assume there is superposition of states (due to external factor or temperature) and this leads to radiation "permanently".