Electron moving in an electromagnetic field and rotation operator

[NiRK20]

Homework Statement

The hamiltonian for an electron in electromagnetic field plus a central potential is
$$H = \frac{1}{2m}\left[ \textbf{p} + \frac{e}{c}\textbf{A}(\textbf{x}, t) \right] ^{2} - e\Phi(\textbf{x},t) - \mu \cdot \textbf{B}(\textbf{x}, t) + V(r)$$
with the gauge $\textbf{A} = \frac{1}{2}\textbf{B}\times \textbf{x}$ and $\Phi = 0$, the field being uniform $\textbf{B} = B \hat{\textbf{b}}$ and $\mu = -\frac{2\mu_{B}}{\hbar}\textbf{S}$.

The problem then gives time-dependinig Schrödinger equation and says to define a new state $| \phi(t) \rangle$ such as
$$| \psi(t) \rangle = U(\hat{\textbf{b}}, \omega t)| \phi(t) \rangle$$
with $U(\hat{\textbf{b}}, \omega t)$ a rotation operator for all degrees of freedom. Finally, it asks to find a frequency $\omega$ that eliminate the effects of the magnetic field over the particle orbital movement except by a centrifuge potental proportional to $(\textbf{b}\times \textbf{x})^{2}$.

Relevant Equations

$$i\hbar \frac{\partial}{\partial t}| \psi(t) \rangle = H | \psi(t) \rangle$$

Hello,

The idea I had was to time evolve the state $U(\hat{\textbf{b}}, \omega t)| \phi(t) \rangle$, but I'm confused on how to operate with $H$ on such state. I Iwould be glad if anyone could point some way. Thanks!