We consider a quantum charged particle moving in the
xy plane under the action of a uniform perpendicular time-dependent magnetic field, in the presence of a parabolic binding potential. The time-dependent probability distribution of the magnetic moment is calculated analytically and numerically in the case of initial thermodynamic equilibrium state. In the high-temperature regime, the initial distribution is almost symmetric, resulting in a tiny mean diamagnetism. However, if the magnetic field eventually changes its sign, the fragile balance between the diamagnetic and paramagnetic ``wings'' of the probability distribution becomes broken, resulting in a giant mean diamagnetic moment, exceeding the initial one by several orders of magnitude. The final mean value of the magnetic moment is inversely proportional to the strength of the binding potential, and it does not depend on the Planck constant in the high-temperature regime. Strong fluctuations of the magnetic moment (described in terms of the variance) exist in all temperature regimes.
Abstract I consider a quantum spinless charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the linear vector potential in the ‘circular’ gauge A = B ( t )(− y , x )/2 and in the presence of the confining potential m g 2 x 2 + y 2 / 2 . This potential guarantees finite values of all second-order moments in the initial equilibrium (thermal) state. Time-dependent mean values of the energy and magnetic moment are expressed in terms of solutions to the classical equation of motion ε ̈ + Ω 2 ( t ) ε = 0 . Simple approximate results are found in the cases of the sudden jump of magnetic field, the parametric resonance, and the adiabatic evolution. They are compared with exact solutions in terms of the hypergeometric function for the specific function B ( t ) of the Epstein–Eckart type. A great amplification of the magnetic moment is discovered for the initial high-temperature states, both for the fast and slow monotonous variations of the magnetic field. Classical adiabatic invariants do not exist if the magnetic field slowly goes to zero or changes its sign.
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 ${\bf A}=B(t)\PG{-y(1+\alpha),x(1-\alpha)}/2$, with two fixed values of the gauge parameter
α:
α=0 (the circular gauge) and
α=1 (the Landau gauge). Although 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 $\ddot\vep +\omega_{\alpha}^2(t) \vep=0$, 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 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.
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.
Landau's diamagnetism for a free point charge is shown to exist within classical electron theory with classical electromagnetic zero-point radiation. The system considered is a nonrelativistic classical point charge bound harmonically in three dimensions and situated in a magnetic field when random classical electromagnetic zero-point radiation or thermal radiation is present. The average energy, angular momentum, and magnetic moment are calculated at finite temperature, and then carried to the limit at which the harmonic binding vanishes so as to obtain the behavior for a free particle. In the presence of the Rayleigh-Jeans radiation spectrum one finds that all diamagnetic effects vanish, in agreement with the results of traditional classical statistical mechanics. In the Planck radiation spectrum one finds exactly the results of quantum theory involving a Langevin function for the temperature dependence. In particular, the average angular momentum and magnetic moment for a classical point charge in classical zero-point radiation are $〈{L}_{z}〉=\ensuremath{-}\ensuremath{\hbar}$ and $〈{M}_{z}〉=\ensuremath{-}|e|\frac{\ensuremath{\hbar}}{2\mathrm{mc}}$, where the orientation of the
z axis is given by the magnetic field direction. Thus the classical results in the presence of classical zero-point radiation are entirely different from those of traditional classical electron theory, and they suggest possible classical explanations of the space quantization appearing in quantum theory. The results also suggest a derivation of Planck's spectrum from traditional classical statistical mechanics by applying Boltzmann statistics to the orientation of the average magnetic moment caused by the zero-point radiation.
We consider N fermionic particles in a harmonic trap initially prepared in a thermal equilibrium state at temperature β^{-1} and examine the probability density function (pdf) of the work done by a magnetic field slowly varying in time. The behavior of the pdf crucially depends on the number of particles N but also on the temperature. At high temperatures (β≪1) the pdf is given by an asymmetric Laplace distribution for a single particle, and for many particles it approaches a Gaussian distribution with variance proportional to N/β(2). At low temperatures the pdf becomes strongly peaked at the center with a variance that still linearly increases with N but exponentially decreases with the temperature. We point out the consequences of these findings for the experimental confirmation of the Jarzynski equality such as the low probability issue at high temperatures and its solution at low temperatures, together with a discussion of the crossover behavior between the two temperature regimes.