###### Abstract
Functions (6) are orthogonal and normalized:  
\[\langle\Psi_{n_{r}^{\prime}m^{\prime}}(t)|\Psi_{n_{r}m}(t)\rangle=\delta_{mm^{ \prime}}\delta_{n_{r}^{\prime}n_{r}}. \tag{10}\]  
Various results related to quantum systems described by means of time-dependent Hamiltonian (2) were found over the past five decades after paper [5]: see, e.g., papers [6; 7; 8; 9; 10; 11; 12; 13; 14]. However, an interesting special case of time inversion of the magnetic field, especially in the adiabatic regime, was not considered in previous studies, despite adiabatic approximation in quantum mechanics being the subject of numerous papers: see, e.g., [15; 16; 17; 18; 19; 20; 21]. The aim of our paper is to fill in this gap (at least partially). Remember that stationary homogeneous magnetic fields can be created inside solenoids with constant currents. On the other hand, alternating currents are quite common in our daily lives. Hence, it can be interesting to understand, what can happen with initial quantum states of a charged particle placed inside a solenoid with an alternating current. It is worth noticing in this connection, that periodic variations of the magnetic field created by a current with the standard frequency \(50\,\mathrm{Hz}\) can be considered as adiabatic, if the amplitude of the magnetic field is sufficiently large. Indeed, the Larmor frequency of an electron in the magnetic field of the order of \(1\,\mathrm{T}\) is of the order of \(10^{11}\,\mathrm{s}^{-1}\). This means that the period of rotation of the particle is many orders of magnitude smaller than the scale of variations of the magnetic field. This presents one of the motivations behind our study. The adiabatic evolution of the unidimensional harmonic oscillator, whose frequency passes through zero, was studied in the recent paper by [PERSON]. [23]. However, when the frequency turns into zero and returns to nonzero values for this system, the situation seems to be rather exceptional (although possible). On the other hand, a passage of the Larmor frequency through zero is quite natural if the magnetic field changes its sign. In addition, the existence of the second degree of freedom and additional invariant (the canonical angular momentum) adds more interesting features to the dynamics of the system described by means of Hamiltonian (2). One of these new features is the evolution of the magnetic moment. Another interesting feature is the infinite degeneracy of the stationary energy levels in the presence of a magnetic field. An analysis of these additional features represents the second motivation of the present study. The remainder of this paper is structured as follows. In Section 2, we bring general formulas describing the adiabatic evolution of the Fock states and the magnetic moment, including the adiabatic regimes with and without crossing zero value of the Larmor frequency. Sections 3 and 4 are devoted to an analysis of the new adiabatic parameters in the cases of single and multiple transitions of the frequency through zero. In Section 5, we consider the violation and generalization of the Born-Fock theorem when a magnetic field slowly changes its sign. Analytical results are illustrated with several figures. In Section 6, we study the evolution of the mean energy and the mean magnetic moment (as well as their fluctuations) in the case of initial Fock states. A more general case of initial \"invariant states\" of the magnetic moment operator is considered in Section 7. Section 8 provides the discussion of main results.