## 5 Generalization of the Born-Fock Theorem  
The time-dependent solution (6) is a superposition of adiabatic eigenstates (3) with the instantaneous value of frequency \(\omega(t)\) and the same value of the canonical angular momentum quantum number \(m\):  
\[\Psi_{n,m}(r,\varphi;t)=\sum_{q_{r}=0}^{\infty}C^{(m)}_{n,q_{r}}(t)\psi_{q,m}( r,\varphi),\qquad C^{(m)}_{n,q_{r}}(t)=\langle\psi_{q,m}|\Psi_{n,m}(t)\rangle. \tag{39}\]  
Coefficients \(C^{(m)}_{n,q_{r}}(t)\) are oscillating functions of the phase \(\phi(t)\) [defined in Equation (21)]. It is remarkable, however, that the modules squared \(|C^{(m)}_{n,q_{r}}|^{2}\) do not oscillate: they depend on the absolute values of constant parameters \(u_{\pm}\) only, as shown in Appendix B:  
\[|C^{(m)}_{n,q_{r}}|^{2}=\frac{(n_{>}+|m|)!n_{<}!|u_{-}|^{2|q_{r}-n_{r}|}}{(n_ {<}+|m|)!n_{>}!|u_{+}|^{2(|q_{r}-n_{r}|+|m|+1)}}\left[p^{(|q_{r}-n_{r}|,|m|)}_ {n_{<}}\bigg{(}\frac{1-|u_{-}|^{2}}{1+|u_{-}|^{2}}\bigg{)}\right]^{2}. \tag{40}\]  
Here, \(P^{(a,b)}_{k}(z)\) is the Jacobi polynomial. Other notations are as follows:  
\[n_{<}=\min(q_{r},n_{r}),\qquad n_{>}=\max(q_{r},n_{r}).\]  
Formula (40) has many interesting consequences. First, it does not depend on the sign of quantum number \(m\), while the energy levels (5) are different for positive and negative values of \(m\). Second, the probabilities (40) are different for different positive values of \(m\), although the initial energy eigenvalues are the same, if \(\omega(t<t_{*})>0\). If \(\omega(t)<0\) for \(t>t_{*}\), eigenstates \(\psi_{q,m}(r,\varphi)\) in the expansion (39) describe the states with the instantaneous energy eigenvalues \(E_{q,m}(\omega<0)=\hbar|\omega(t)|(1+|m|+m+2q_{r})\), which are different for different positive values of \(m\), in contradistinction to the infinite energy levels degeneracy at \(t<t_{*}\). The simplest special case is \(n_{r}=0\), when \(n_{<}=0\) and \(n_{>}=q_{r}\). Then, the Jacobi polynomial turns into unity, so that  
\[|C^{(m)}_{0q_{r}}|^{2}=\frac{(q_{r}+|m|)!|u_{-}|^{2q_{r}}}{|m|!q_{r}!|u_{+}|^{2 (q_{r}+|m|+1)}}. \tag{41}\]  
One can verify that \(\sum_{q_{r}=0}^{\infty}|C^{(m)}_{0q_{r}}|^{2}=1\), due to the formula  
\[S_{m}(x)=\sum_{k=0}^{\infty}\frac{(m+k)!}{m!k!}x^{k}=(1-x)^{-m-1}. \tag{42}\]  
The mean energy in the distribution (41) can be easily calculated with the aid of Formula (42):  
\[\langle\hat{H}(t)\rangle = \hbar|\omega(t)|\sum_{q_{r}=0}^{\infty}|C^{(m)}_{0q_{r}}|^{2} \bigg{(}1+|m|-\frac{\omega(t)}{|\omega(t)|}m+2q_{r}\bigg{)} \tag{43}\] \[= \hbar|\omega(t)|\bigg{[}1+|m|-\frac{\omega(t)}{|\omega(t)|}m+2(1 +|m|)|u_{-}|^{2}\bigg{]}.\]Remember that the initial energy was \(E_{in}=\hbar|\omega_{in}|(1+|m|-m)\), and all initial states with \(m\geq 0\) had the same energy. However, when the Larmor frequency passes through zero and changes its sign, this infinite degeneracy is removed. Moreover, the degeneracy is removed even when the frequency maintains its sign, if \(u_{-}\
eq 0\). In any case, the adiabatic ratio \(\langle\hat{H}(t)\rangle/|\omega(t)|\)_always_ increases when the frequency passes through zero. The distribution (41) as a function of \(q_{r}\) has a rather simple form. It decays monotonously if \(m=0\), going to a distribution with a single maximum at \(q_{r}+1\approx|m|\,|u_{-}|^{2}\) for large values of \(|m|\). However, the situation is more intricate for nonzero values of the initial radial quantum number \(n_{r}=n\). In the special case when \(|u_{-}|^{2}=1\) and \(|u_{+}|^{2}=2\) (a single linear transition of the magnetic field through zero), the general distribution (40) assumes the form  
\[|C_{nq}^{(m)}|_{|u_{-}|=1}^{2} = \frac{(n_{>}+|m|)!n_{<}!}{(n_{<}+|m|)!n_{>}!2|q^{-n|+|m|+1}}\Big{[} \!P_{n_{<}}^{(|q-n|,|m|)}(0)\Big{]}^{2} \tag{44}\] \[= \frac{(n+|m|)!(q+|m|)!n!q!}{2^{n+q+|m|+1}}\Bigg{[}\!\sum_{k=0}^{n _{<}}\frac{(-1)^{k}}{k!(n-k)!(q-k)!(k+|m|)!}\Bigg{]}^{2}.\]  
Several examples of the distribution (44) as a function of \(q_{r}=q\) are shown in Figures 1 and 2 for various values of parameters \(n_{r}=n\) and \(|m|\).