So, the Semiclassical Model just uses classical equations of motion, of a particle hbar/k to calculate electron trajectories. We then make the assertion that these trajectories are just the trajectories of the wave packets, so we're not violating uncertainty. I was wondering; can anyone think of a way to describe the electrons probabilistically, without something horrendous?
Edit:
I just thought I'd clarify: I mean, rather than calculating wave packet trajectories, is there a more general approach? I can't think of any that wouldn't be horrible to work with. I guess what I'm saying is that the semiclassical approach is quite nice...
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I would just like to comment that semi-classical models whereby we treat the particles quantum mechanically as waves and external fields as classical fields has been popping up a lot lately in the courses we are doing. As for your question, since the validity of the semi-classical approach breaks down for high E and B fields, I am sure there is a more general approach which probably involves quantisation of the E and B fields a.k.a. QFT. But is really nice that semi-classical model works well in the regimes we are usually interested.
ReplyDeleteYeah, this is starting to sound like our work in Quantum...
ReplyDeleteI suppose you could even take a sort of stochastic approach and brute force many simulations for single particles and take an average? :p
This semiclassical stuff reminds me of the Gross–Pitaevskii equation...
From what I can gather about the semiclassical model is that one now deals with crystal momentum rather than the physical momentum of mass x velocity. So whilst previously we defined hbar*k as momentum, now it takes on this new crystal momentum value.
ReplyDeleteThis value is related to momentum, as it is related to the kinetic energy of the particle (which we say we know) by the usual relation (hbar*k)^2/2m. However, the proper momentum is uncertain, due to the uncertainty principle, so QM isn't totally violated or ignored. That's the interpretation I've taken so far, but could very easily be wrong there...
Yeah, they are the best equations that do a reasonable good job at describing the system while also giving you a bit of a gut feeling of what is happening - i.e. you can see the what forces are important etc. If you wanted to go a full quantum approach you would have to solve a SE like the one we saw today in class which would be kinda good. I think maybe doing a many-body problem in the interaction picture would be much nicer (still probably quite horrible) than an direct wave function approach
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