What is the External Field Approximation in Quantum Mechanics?

[Miquelyn10]

This is not a homework question I'm just trying to understand quantum mechanics. I have found a Hamiltonian that has the potential engery part it as [eFr cos(wt)]. All of the variables are known but I can't identify what F is. It just states that F denotes the amplitude of the external field.

I started do so some research and found out there is something called External Field Approximation. I can't really find what that is. Is there a formula to find the external field approximation? Any help would be greatyl appreciated!

 

[Miquelyn10]

Can anybody please help me?

 

[Mike Pemulis]

I don't think it's very mysterious or anything. Your Hamiltonian describes a harmonically varying classical electric field incident on an electron or other charged particle. So yeah, F is just the amplitude (strength) of the external electric field. The value of F depends on the experimental apparatus that is creating the field -- it's not a physical constant or anything.

The reason that this is an approximation is that we are treating the field as a classical entity. But of course the universe is actually quantum-mechanical, so we should describe the field, as well as the electron, quantum-mechanically. However, the classical approximation works very well for macroscopic field amplitudes.

In concrete terms, if your experiment is a laser shining on an electron, you can treat the laser light as a classical EM field, because the light is composed of so many photons that the individual photons are not important. But if your experiment is a single photon interacting with the electron, the photon is not a classical field anymore but a quantum field.

Notice that in both experiments, the electron is quantum-mechanical because it is just one particle. Turning on quantization of the electric field as well has been called "second quantization" although I think that term is deprecated now.

Edit: thought of something else. The "External Field Approximation" can also refer to the assumption that the electric field is not affected by the motion of the electron (although we know that moving charges create fields also). So the applied field is just moving the electron around; the electron does not create a field of its own. Hence, External Field.

 

[Miquelyn10]

Thanks for the explanation but how do I go about actually finding the amitude of te external field?

 

[SpectraCat]

Thanks for the explanation but how do I go about actually finding the amitude of te external field?

You can either take it as arbitrary, or else it will be determined by the parameters of a specific case.