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

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

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.