I have been doing a lot of exercises out of Griffiths lately, and one thing I spent a lot of time thinking about was Larmor precession. It's a fun thing to work out, but where does it actually happen? I'm having a hard time to see where it would apply.(adsbygoogle = window.adsbygoogle || []).push({});

The first condition for Larmor precession to occur is that you have a spin-1/2 particleat rest. First off, the at rest part bugs me in part because it seems impossibly difficult to have a free electron "at rest" relative to your lab. In theory, I get that it works out. "At rest" in this context means that ##\langle \vec{p} \rangle = 0##, which is well and good, since it implies that ##\langle \vec{p} \cdot \vec{A} \rangle = 0##. This means there should be no canonical momentum funky business coming into the Pauli equation, and so the particle should look like a standing wavepacket, leaving only the spin-y stuff affecting the Hamiltonian. What bugs me is that it seems impossibly difficult to get a free electron at rest in the lab frame. For starters, you would need to do the experiment in a Faraday cage to keep out the air waves, but with the walls far from the electrons so they don't have a mirror charge reaction, which would make the electron move. Earnshaw's theorem would suggest that this experiment can't be done without something forcing the electron to stay at rest. You can't optically trap a free electron, because there is no restoring force, so your laser will always be blue detuned off resonance no matter what. I suppose the next best thing to a free electron is a conduction electron in a material, but common sense says that if you put a magnetic field in a conductor it heats up because of eddy currents: moving electrons. That empirical fact would seem to suggest that conduction band electrons don't satisfy ##\langle \vec{p} \rangle = 0##, I suspect due to thermal motion. Say you got it down to 1.8K with a helium cryo, assuming the conductor doesn't start superconducting, then maybe you could get the electrons to precess a little before they start gaining momentum. Still, I highly doubt that we care about Larmor precession for that application, if it even worked.

I have a limited understanding of these things, but I suspect that the real reason we care about Larmor precession is that the behavior of an atom in a magnetic field can be decomposed into an atomic part describing atom-y stuff (Zeeman splitting of orbits, etc.) and a part describing the precession of the electron as it "orbits" the nucleus, via an interaction picture or some other fancy transformation of the Pauli equation Hamiltonian. This seems a lot more important an application from an atomic physics point of view. Am I getting warmer? If so, I would actually like to try this problem. Can I get out the Larmor precession Hamiltonian as an interaction term starting with the Pauli equation for a hydrogen atom in a magnetic field? If this is not the case, please let me know so I don't sink a bunch of hours into it.

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# I Where Does Larmor Precession Occur?

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