Quantum electron and field interactions

Question

What is the proper way to consider the electric field generated by an electron wavefunction governed by the Schrodinger equation? Can you get a result that would match observation, or is this a shortcoming of first quantization requiring some more advanced field theory?

I have two pictures in mind (though it's not necessary that either are correct):

One could calculate the field from the Lienard-Wiechert potential of a moving point charge, using the expectation values of the electron momentum to give $\mathbf v(t)$.

Alternatively, if one treats the electron charge distributed with the probability density, one could consider a current density $\rho(\mathbf x,t)$ and calculate the field from it's time evolution.

While my intuition suggests the latter makes more sense, this is a description of an interaction between the electron and photons, which would collapse the wavefunction (right?). I'm entirely open to the idea that neither picture is accurate and one needs to reframe the problem altogether.

Answer 1

this is a description of an interaction between the electron and photons, which would collapse the wavefunction (right?).

No this isn't right. As long as the system stays isolated, the interaction simply means that there are cross terms in the relevant Hamiltonian and that you have a two-particle quantum system, whose state space is the tensor product of the two individual one-particle state spaces. The quantum state in this tensor product space then evolves unitarily. However, if you insist on looking only at the state of, say, the electron alone, then it will not evolve unitarily and will look somewhat like a classical mixture described by a density matrix, so in this sense there is a vague likeness between this picture and the Wigner friend thought experiment wherein we know a measurement has been made on the electron, but we do not know the measurement result.

The full picture, with electrons interacting with photons and loops of all orders, is the full QED picture. An interesting intermediate picture, somewhat like you're thinking with your distributed charge idea, is the Nonlinear Dirac equation or the Maxwell-Dirac equation wherein one electron interacts with the EM field, as described in my answer here. In this picture, the "four current" "driving" the EM field is $q\,\bar{\psi} \gamma^\mu \psi$ where $\psi$ is the Dirac spinor electron wavefunction. So this is more like your distributed charge model, although it includes charge flux as well to get a full four current. The Maxwell equations are then $\partial_\mu A_\nu - \partial_\nu A_\mu=q\,\bar{\psi} \gamma^\mu \psi$ and then the four-potential then couples back into the electron field through the Dirac equation $\gamma^\mu\left(i \partial_\mu - q A_\mu\right) \psi + V \psi - \psi = 0$.

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Further Questions from OP

aha - you're saying that if you include the photon(s) in the system from the start, there is no "collapsing". Are you suggesting that the wavefunction collapsing caused by a measurement event is just some handwaving way of describing an interaction with a particle that is not included in the Hamiltonian/system?

As you know, there isn't an accepted full description of the quantum measurement problem yet, but yes, its possible (even likely IMO) that this kind of idea is what explains measurement and that all measurement would become unitary evolution if we included the "measurer" properly in the system description. User DanielSank has more to say about this in his wonderful answer here. Another enticing notion is the notion of a Purification of a Quantum State: that every mixed state can be thought of as a reduced state in a larger system.

Also: do you have any idea how using the Schrodinger electron wavefunction as a source in Maxwell's equations would compare with the Dirac Equation + Maxwell's eq. as you suggest? I mean, is the former fundamentally wrong for some reason, or just not as complete of a picture as considering the Dirac Equation?

I don't think there is anything fundamentally wrong with the notion of coupling the nonrelativistic Schrödinger equation in the way you suggest; after all, people do speak of minimal coupling between the EM field and the NRSE and you get the same notion of gauge covariant derivative as you would with the Dirac equation. It's simply that the NRSE is too inaccurate to model the effects that the study of coupling lets you understand. The effects you model here are relativistic - you yourself are already talking about Liénard–Wiechert potentials - and the Dirac equation replaced the NRSE to overcome this limitation. It gave spectra that were much more in keeping with experiment than the NRSE. The effects of coupling are much smaller than the errors that one corrects in going from NRSE to Dirac - the most important one is the Lamb shift and this splits eigenstates that are degenerate in the Dirac model, so you wouldn't go back to a more inaccurate model before adding the correction!

Answer 2

Let's look at where the electromagnetic interaction comes from in hydrogen.

At first quantization you have a multiparticle system so the wavefunction is defined as $\psi=\psi(x_1,y_1,z_1,x_2,y_2,z_2,t)$ and the point is to write the Hamiltonian.

And the Hamiltonian comes from the Lagrangian. For a single particle of charge $q$ in an external electromagnetic potential the Lagrangian is $L(\vec x, \vec v)=-mc^2\sqrt{1-\frac{v^2}{c^2}}+q\vec A\cdot\vec v -q\Phi.$ Which leads to the Hamiltonian $H(\vec Q,\vec P,t)=\sqrt{(mc^2)^2+(\vec P-q\vec A)^2c^2}+q\Phi$ (which is approximately $mc^2 +(\vec P-q\vec A)^2/2m +q\Phi$ in the nonrelativistic limit).

If your goal is to find energy eigenstates you might want to consider the quantum versions of equilibrium and an isolated system so the center of mass of the system moves inertially and the two charged bodies exert roughly equal forces and so the $\approx$ 2000 times as massive proton moves pretty closely to inertially so in its frame the potential is all scalar and that goes inversely to $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}.$

This naturally suggests a change of variables where the two particle configuration is changed to a center of mass location and a relative position of the electron. The point of all this is that this is the kind of argument usually used to get a Hamiltonian that is a sum of a free particle (all kinetic) of the center of mass and a $P^2/2\mu -|e|/r$ for the relative position $\vec r$ and reduced mass $\mu.$

So the point is that you can see how things interact electromagnetically if you write the Lagrangian for all the charges, then do a Legendre transformation to get the Hamiltonian, then use that Hamiltonian or a nonrelativistic approximation (since electromagnetism is naturally already relativistic).

That is how it is done. And that's the first stab at an answer. Showing you how it is normally done, so you know the traditional approach for a Schrödinger equation level electromagnetic interaction.

So you generally need that Lagrangian for the electromagnetic interaction between the charges. But you want the Lagrangian to depend on the configuration and the time rate of change of position. But in general there isn't an obvious potential of one charge on the other that only depends on the current positions and velocities of the particles, there should be a time delay, you find the potential on the past light cone. But if you imagine the non relativistic limit as a high $c$ limit you could choose to use a wave equation for the potential with charge and current at the present time as the source. In the realm of nonrelativistic approximations the most common techniques for non experts is to either make something up and hope it is good or do an order by order (in v/c) expression for the system and truncate after some order.

An example system you can look at would be $H^+$, a proton with two electrons at which point you can ask yourself how the two electrons interact with each other electromagnetically. You can write a Lagrangian, get a Hamiltonian, approximate them to some nonrelativistic approximation, and then set up the quantum system. If you have two electrons you will have use the superselection rule, and will probably need to include spin to do the super selection (anti symmetrize the wavefunction under exchange of identical fermions).

And that's unfortunate, since it means you don't start with a known classical Lagrangian for a system with spin degrees of freedom. A most basic version could be to treat the spin as a magnetic dipole that is proportional to the spin, and include the terms in the Lagrangian corresponding to an interaction between a magnetic dipole and a vector potential.

OK, that's an attempt at an honest recap of how electromagnetic interactions are often dealt with in regular Schrödinger equation level first quantization. Sometimes people have external fields, do perturbation, etcetera. However you asked about a quantum source not an external source on a quantum feeler.

If you want to see an electromagnetic field outright, you want that to be an object in its own right. For that, at the classical level the potential itself is part of the configuration in addition to the locations of the charges. But when you write the Lagrangian for that it is a Lagrangian density and when it gets quantized you get quantum field theory which you said isn't what you want.

So that's another answer. To get a true electromagnetic field you need quantum field theory. If all you want is the electromagnetic effect of the quantum charged object then you can do it the same way you normally do, which I've outlined above.