In the French wiki article about the Ehrenfest theorem I found these formulas.

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {x}}\rangle ={\frac {1}{m}}\langle {\hat {p}}\rangle }$$ and

$${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\langle {\hat {p}}\rangle =\langle F\rangle }.$$

I consider the quantum problem of an electron in a constant magnetic fiels B. the generalized momentum is $P = p - qA$ where the vector potential is written in the Landau gauge: $(0,Bx,0)$ and the Hamiltonian is $H = P^2 / 2m$.

How can I use the Ehrenfest theorem to calculate the classical radius of the circle in the x,y plane given by the classical theory?

the Landau gauge is described here


1 Answer 1


A rather straightforward approach is tow ork in the Heisenberg picture: write the equations-of-motion for operators using the general prescription $$ \dot{A}=\frac{1}{i\hbar}[A,H]_-,$$ and average them over the initial state.

One however obtaines this result in an almost identical way using the Schrödinger picture, operating with operators of derivatives (rather than the time derivatives of operators in Heisenberg picture, see here).

  • $\begingroup$ Your answer is correct, but it is too general. my question is about a pecular gauge and how to get the circles with Ehrenfest theorem. $\endgroup$
    – Naima
    Nov 26, 2021 at 8:38
  • $\begingroup$ @Naima I think the necessary derivations are quite simple. I don't see a conceptual issue here, and we are discourage from posting homework solutions here. Perhaps, you could show in your question how you try to solve it and where you get stuck. $\endgroup$
    – Roger V.
    Nov 26, 2021 at 9:15

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