Abstract
We consider a quantum spinless nonrelativistic charged particle moving in the plane under the action of a time-dependent magnetic field, described by means of the linear vector potential , with two fixed values of the gauge parameter : (the circular gauge) and (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 and straight lines for . 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 . They are expressed in terms of solutions to the classical equation of motion , with . 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 , 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.
Highlights
The motion of a quantum charged particle in a uniform stationary magnetic field has attracted the attention of many authors since the first years of quantum mechanics [1,2,3,4,5,6]
As soon as we are interested in the evolution of the mean energy and mean magnetic moment, we have to calculate the mean values of various products of operators xr,c and yr,c as functions of time
We have obtained several exact results describing the dynamics governed by Hamiltonian (1) with two gauges: the circular and Landau ones
Summary
The motion of a quantum charged particle in a uniform stationary magnetic field has attracted the attention of many authors since the first years of quantum mechanics [1,2,3,4,5,6]. That no explicit solutions to Equation (3) with ωα(t) = const were considered in all the cited papers It was mentioned already in paper [10], that the physical consequences are different for the two gauges in the time-dependent magnetic fields. The goal of our paper is to compare the explicit evolution of such physical quantities as the mean energy and mean magnetic moment, as well as their variances, for two different physical systems, characterized by two different gauge parameters of the time-dependent vector potential with the same magnetic field B(t). In addition to the energy, there exists another quadratic integral of motion, which can be considered as the generalized angular momentum (the same formulas hold for the classical variables and quantum operators): L It coincides formally with the canonical angular momentum Lcan = xpy − ypx in the only case of “circular” gauge of the vector potential. Another proof of the definition (12) for an arbitrary gauge was given in [56] (see [57,58,59,60,61])
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
