Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids

Abstract

We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential A=B(t)−y(1+α),x(1−α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.

Adiabatic Amplification of Energy and Magnetic Moment of a Charged Particle after the Magnetic Field Inversion

“…Some results presented in the paper by Ref. [ 36 ] indicate that the evolution can be quite different for the Landau gauge, because of another geometry of the induced electric field. However, the general linear gauge of the time-dependent magnetic field remains an unsolved problem.…”

“…If , the calculations become rather cumbersome, so we do not perform them here. For explicit expressions in some special cases, one can consult papers [ 32 , 36 ].…”

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

“…The squeezing operator is the generator of the Bogoljubov transformation, which has played an important role in the theory of superfluidity (Bogoljubov 1947, Dodonov 2007). The quantum dynamics of charged particles in time-varying external (electro)magnetic fields also shows interesting phenomena, related to non-classical states of linear oscillators (Malkin et al 1970, Varró 1984, Varró and Ehlotzky 1985, Dodonov 2018, Dodonov and Horovits 2021. In the context of Landau states, concerning coherent states and non-classicality, see the detailed discussions by Fakhri, Mojaveri and Gomshi Nobary (2010), Dehghani et al (2012), Dehghani and Mojaveri (2013) and Mojaveri and Dehghani (2015).…”

Coherent and incoherent superposition of transition matrix elements of the squeezing operator

“…The squeezing operator is the generator of the Bogoljubov transformation, which has played an important role in the theory of superconductivity (Bogoljubov 1958, Valatin 1958). The quantum dynamics of charged particles in time-varying external (electro)magnetic fields also shows interesting phenomena, related to non-classical states of linear oscillators (Malkin, Man'ko and Trifonov 1970, Varró 1984, Varró and Ehlotzky 1985, Dodonov 2018, Dodonov and Horovits 2021. Recently the interest in entangled states is more pronounced then in nonclassical pure states.…”