Abstract
We consider a quantum charged particle moving in the x y plane under the action of a time-dependent magnetic field described by means of the linear vector potential of the form A = B ( t ) − y ( 1 + β ) , x ( 1 − β ) / 2 . Such potentials with β ≠ 0 exist inside infinite solenoids with non-circular cross sections. The systems with different values of β are not equivalent for nonstationary magnetic fields or time-dependent parameters β ( t ) , due to different structures of induced electric fields. Using the approximation of the stepwise variations of parameters, we obtain explicit formulas describing the change of the mean energy and magnetic moment. The generation of squeezing with respect to the relative and guiding center coordinates is also studied. The change of magnetic moment can be twice bigger for the Landau gauge than for the circular gauge, and this change can happen without any change of the angular momentum. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields.
Highlights
The motion of a quantum non-relativistic charged particle in a uniform stationary magnetic field, B = (0, 0, B) = rot A, has been studied since the dawn of quantum mechanics, beginning with the papers by Kennard, Darwin, and Fock [1,2,3]
The Schrödinger equation in the case of time-dependent uniform magnetic field was solved by Malkin, Man’ko, and Trifonov [5] for the “circular” gauge of the vector potential and by Dodonov, Malkin, and Man’ko for the Landau gauge [6]
M = −eE /(mcω0), for any definition of the magnetic moment operator. This relation explains why different choices of the vector potential gauge do not influence the final results for the magnetization of the free electron gas in the time-independent magnetic field
Summary
The Schrödinger equation in the case of time-dependent uniform magnetic field was solved by Malkin, Man’ko, and Trifonov [5] for the “circular” gauge of the vector potential and by Dodonov, Malkin, and Man’ko for the Landau gauge [6]. In these two cases, the physical results turned out quite different, e.g., comparing the transition probabilities between the energy levels.
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