###### Abstract  
We study the evolution of the energy and magnetic moment of a quantum charged particle placed in a homogeneous magnetic field, when this field changes its sign adiabatically. We show that after a single magnetic field passage through zero value, the famous adiabatic invariant ratio of energy to frequency is reestablished again, but with a proportionality coefficient higher than in the initial state. The concrete value of this proportionality coefficient depends on the power index of the frequency dependence on time near zero point. In particular, the adiabatic ratio of the initial ground state (with zero radial and angular quantum numbers) triplicates if the frequency tends to zero linearly as a function of time. If the Larmor frequency attains zero more than once, the adiabatic proportionality coefficient strongly depends on the lengths of the time intervals between zero points, so that the mean energy behavior can be quasi-stochastic after many passages through zero value. The original Born-Fock adiabatic theorem does not work after the frequency passes through zero. However, its generalization is found: the initial Fock state becomes a wide superposition of many instantaneous Fock states, whose weights do not depend on time in the new adiabatic regime. generalized adiabatic invariants; generalized Born-Fock theorem; time-dependent Larmor frequency; mean energy; energy fluctuations; mean value and fluctuations of magnetic moment; single and multiple frequency passages through zero 2023259660  
\[\psi_{n_{r}m}(r,\varphi)=\sqrt{\frac{\kappa n_{r}!}{\pi(n_{r}+|m|)!}}\Big{(}\kappa r ^{2}\Big{)}^{|m|/2}L_{n_{r}}^{(|m|)}\Big{(}\kappa r^{2}\Big{)}\exp\Bigl{(}- \kappa r^{2}/2+im\varphi\Bigr{)}, \tag{3}\]  
\[\kappa=|\omega|/\hbar,\qquad m=0,\pm 1,\pm 2,\ldots,\quad n_{r}=0,1,2,\ldots. \tag{4}\]  
Function \(L_{n}^{(\alpha)}(z)\) is the generalized Laguerre polynomial, defined as [3; 4]  
\[L_{n}^{(\alpha)}(z)=\frac{1}{n!}\varepsilon^{z}z^{-\alpha}\frac{d^{n}}{dz^{n} }\bigl{(}e^{-z}z^{\alpha+n}\bigr{)}.\]  
The energy eigenvalues are given by the formula  
\[E_{n_{r}m}=\hbar|\omega|(1+|m|+2n_{r})-\hbar\omega m, \tag{5}\]  
which consists of two parts, in accordance with the two parts of Hamiltonian (2). Note that the radial part of the wave function (3) depends on the absolute value \(|m|\) of the canonical angular momentum eigenvalue. The algebraic value \(m\) (which can be either positive or negative) enters through the preserved canonical angular momentum operator in Hamiltonian (2). If \(\omega>0\), all states with \(m\geq 0\) have the same energy \(\hbar|\omega|(1+2n_{r})\), meaning an infinite degeneracy of these energy levels. On the other hand, the same infinite degeneracy happens for all states with \(m\leq 0\) if \(\omega<0\). These observations are important for the interpretation of results in the following sections. What happens with the solution (3), when the magnetic field depends on time after some initial time instant \(t_{i}\)? A general answer was given many years ago by [PERSON], [PERSON] and [PERSON] [5]:  
\[\Psi_{n_{r}m}(r,\varphi;t)=\sqrt{\frac{K(t)n_{r}![K(t)r^{2}]^{|m|}}{\pi(n_{r}+ |m|)!}}L_{n_{r}}^{(|m|)}\Bigl{[}K(t)r^{2}\Bigr{]}\exp\Bigl{(}i\frac{\delta}{2 \hbar\epsilon}r^{2}+im\varphi+i\chi(t)\Bigr{)}. \tag{6}\]  
The real function \(K(t)=\hbar^{-1}|\varepsilon(t)|^{-2}\) is determined by the complex function \(\varepsilon(t)\), which is the solution to the classical equation of the harmonic oscillator with a time-dependent frequency,  
\[\xi+\omega^{2}(t)\varepsilon=0, \tag{7}\]  
satisfying the initial conditions  
\[\varepsilon(t_{i})=[\omega(t_{i})]^{-1/2},\quad\dot{\varepsilon}(t_{i})=i[ \omega(t_{i})]^{1/2}. \tag{8}\]  
These conditions fix the value of the time-independent Wronskian  
\[\dot{\varepsilon}\epsilon^{*}-\dot{\varepsilon}\epsilon=2i. \tag{9}\]  
We assume that \(\omega(t_{i})>0\). Since the phase \(\chi(t)\) is not important for our purposes, we do not provide herein its explicit (rather complicated) expression.