## 6 Mean Energy  
To find the mean energy after the inversion of the magnetic field for an arbitrary initial energy eigenstate, one could try to calculate the sum  
\[\sum_{q_{r}=0}^{\infty}|C^{(m)}_{n,q_{r}}|^{2}\bigg{(}1+|m|-\frac{\omega(t)}{| \omega(t)|}m+2q_{r}\bigg{)}=1+|m|-\frac{\omega(t)}{|\omega(t)|}m+2\sum_{q_{r}= 0}^{\infty}|C^{(m)}_{n,q_{r}}|^{2}q_{r}. \tag{46}\]  
However, this method is not easy, because the sum contains squares of the Jacobi polynomials with variable lower and upper indexes. A direct calculation of the kinetic energy mean value \(\langle\hat{p}_{x}^{2}+\hat{p}_{y}^{2}\rangle/2\) requires a knowledge of complicated integrals containing derivatives of function (6). It is probable that the simplest way is to notice that Hamiltonian (2) is quadratic with respect to the coordinates and momenta operators. Consequently, the Heisenberg or Ehrenfest equations for these operators are linear. This means that their time-dependent mean values are linear combinations of the initial values with certain time-dependent coefficients. It is convenient to introduce vector \(\mathbf{Q}=(x,y,p_{x},p_{y})\) (whose components are either mean values of quantum operators or classical variables). Then,  
\[\mathbf{Q}(t)=\Lambda(t;t_{i})\mathbf{Q}(t_{i}), \tag{47}\]  
where \(\Lambda(t;t_{i})\) is a \(4\times 4\) matrix. Combining mean values \(Q_{jk}\equiv\langle\hat{Q}_{j}\hat{Q}_{k}+\hat{Q}_{k}\hat{Q}_{j}\rangle/2\) into the \(4\times 4\) symmetric matrix \(\mathcal{Q}\), one can verify the relation (see [35])  
\[\mathcal{Q}(t)=\Lambda(t;t_{i})\mathcal{Q}(t_{i})\tilde{\Lambda}(t;t_{i}), \tag{48}\]  
where \(\tilde{\Lambda}\) means the transposed matrix. In our case, matrix \(\Lambda(t;t_{i})\) can be expressed in terms of the solution \(\varepsilon(t)\) and its derivative \(\dot{\varepsilon}(t)\) as follows [34],  
\[\Lambda(t;t_{i})=\omega_{i}^{1/2}\left\| \begin{array}{cc}\mathrm{Re}( \varepsilon)\mathbf{R}&\mathrm{Im}(\varepsilon)\mathbf{R}/\omega_{i}\\ \mathrm{Re}(\varepsilon)\mathbf{R}&\mathrm{Im}(\varepsilon)\mathbf{R}/\omega_ {i}\end{array} \right\|,\qquad\mathbf{R}=\left\| \begin{array}{cc}\cos\tilde {\phi}&\sin\tilde{\phi}\\ -\sin\tilde{\phi}&\cos\tilde{\phi}\end{array} \right\|, \tag{49}\]  
with the phase \(\tilde{\phi}\) defined in Equation (11). Comparing Equations (2), (5) and (15), we obtain the initial mean value  
\[\langle\hat{p}_{x}^{2}+\hat{p}_{y}^{2}\rangle_{n,m}=\omega_{i}(2n_{r}+|m|+1).\]  
Moreover, since the initial wave function (3) possesses rotational symmetry, the initial covariance matrix \(\mathcal{Q}(t_{i})\) can be written in the following block form:  
\[\mathcal{Q}(t_{i})=\frac{\hbar}{2}\left\| \begin{array}{cc}I\gamma/\omega_{i }&m\Sigma\\ -m\Sigma&I\gamma\omega_{i}\end{array} \right\|,\;\;I=\left\| \begin{array}{cc}1 &0\\ 0&1\end{array} \right\|,\;\;\Sigma=\left\| \begin{array}{cc}0&1\\ -1&0\end{array} \right\|, \tag{50}\]  
where \(\gamma=2n_{r}+|m|+1\). In this case, we obtain  
\[\mathcal{Q}(t)=\frac{\hbar}{2}\left\| \begin{array}{cc}I\gamma|\varepsilon(t )|^{2}&m\Sigma+I\gamma\mathrm{Re}(\dot{\varepsilon}\varepsilon^{*})\\ -m\Sigma+I\gamma\mathrm{Re}(\dot{\varepsilon}\varepsilon^{*})&I\gamma|\dot{ \varepsilon}(t)|^{2}\end{array} \right\|. \tag{51}\]  
Consequently,  
\[\langle\hat{H}(t)\rangle=\frac{\hbar\gamma}{2}\Big{[}|\varepsilon(t)|^{2}+| \varepsilon(t)|^{2}\omega^{2}(t)\Big{]}-m\hbar\omega(t). \tag{52}\]  
In the adiabatic regime, using Equation (21), we obtain the formula  
\[\langle\hat{H}(t)\rangle=\hbar|\omega(t)|(2n_{r}+|m|+1)\Big{(}|u_{+}|^{2}+|u_{ -}|^{2}\Big{)}-\hbar\omega(t)m, \tag{53}\]  
which goes to (43) if \(n_{r}=0\). Comparing Equations (24) and (25) with Equation (53), we see that Equation (19) can be generalized as follows,  
\[\overline{\langle\mathcal{M}\rangle}=-\langle\hat{H}(t)\rangle/\big{[}\hbar \omega(t)\big{]}, \tag{54}\]where the overline means an additional time averaging over fast oscillations of the mean magnetic moment, as soon as \(\overline{w(t)}=|u_{+}|^{2}+|u_{-}|^{2}\).