Probability current versus electric current

[BeauGeste]

are these different quantum mechanically?

I thought they were the same since probability current density obeys the continuity equation as the electric current density must also.

prob. current density: ~ psi* grad psi - psi grad psi*
electric current density ~ <psi| p |psi>

are they the same?

 

[PRB147]

They are different, boundary condition is required in the latter case.
The latter case is not dependent on position, while the former is dependent on position.
They are different in the physical significance.

 

[BeauGeste]

thanks. that helps.

 

[Parlyne]

You could have a probability current when dealing with an electrically neutral body. So, no, they're not really the same thing.

 

[cesiumfrog]

You could have a probability current when dealing with an electrically neutral body. So, no, they're not really the same thing.

Huh? Can't you also have an electric current in an electrically neutral body?

 

[Parlyne]

I meant to say you can have a probability current when doing quantum mechanics with uncharged objects like neutrons or neutrinos.

 

[reilly]

are these different quantum mechanically?

I thought they were the same since probability current density obeys the continuity equation as the electric current density must also.

prob. current density: ~ psi* grad psi - psi grad psi*
electric current density ~ <psi| p |psi>

are they the same?

Yes, or course they are, apart from coupling constants. It's an old idea -- say for a problem concerning ionization of a gas illuminated by strong radiation. It's basic to QM, in particular, to suppose that a charged particle with mass and charge will have identical distributions for mass and charge.-- the particle's mass and charge are always at the same place. It started as a "what else could electric current be?" other than the probability current. (Naturally you multiply the probability current by the particle's charge)

There are formal proofs, usually for relativistic QM, based on symmetries, Lorentz invariance -- spin plays a big role. Anyway, such efforts lead to the most general forms for a conserved current, and there's only one possible conserved current for any spin -- there can be a few wrinkles, which I've not mentioned. Most QFT and QED books will at least comment on this equality of currents.

Regards,
Reilly Atkinson