\ll ||H_{\rm O}||$, in which Landau levels change more rapidly than Landau energies, and (iii) the nonadiabatic change when $||H_{\rm D}|| \approx ||H_{\rm O}||$, in which the change of Landau states is comparable to the dynamical phase of Landau energies. The exact quantum motion may not be found for general time-dependent magnetic fields, except for some limiting cases such as a constant magnetic field and a suddenly changing field from one constant value to another. Hence, we propose another scheme which applies a perturbation theory to the Hamiltonian (\[B-tham\]), by treating $H_{\rm D}$ as the unperturbed Hamiltonian and $H_{\rm O}$ as the perturbed one when $||H_{\rm D}|| \gg ||H_{\rm O}||$ or vice versa.

From Eq. (\[ln eq\]), the pair of quantum invariants (\[q-inv\]) in the extended phase space (\[phase\]) $$\begin{aligned}
\hat{I}_{(\pm)} (t) = \frac{1}{\sqrt{2}} \Bigl[ U_{_{(\pm)} \alpha \beta} (\hat{\pi}^*_{\beta} + \hat{\pi}_{\beta}) + V_{(\pm) \alpha \beta} (\hat{\phi}^*_{\beta} + \hat{\phi}_{\beta}) \Bigr], \label{q-inv2}\end{aligned}$$ may be found by solving the matrix-valued differential equations $$\begin{aligned}
\dot{U}_{(\pm)} + U_{(\pm)}  \Omega + V_{(\pm)} = 0, \quad \dot{V}_{(\pm)} - U_{(\pm)} {\bf \omega}^2 + V_{(\pm)} \Omega = 0. \label{uv eq}\end{aligned}$$ Here, the positive (negative) sign denotes the positive (negative) frequency solution. The two equations (\[uv eq\]) are equal to 