Ehrenfest theorem: On which classical circle can we find the electrons in an homogenous magnetic field?

Question

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

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).