The adiabatic motion of a charged, spinning, quantum particle in a two-dimensional (i.e. of constant direction) magnetic field is studied. A suitable set of operators generalizing the kinematical momenta and the guiding centre operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a fast and a slow one which are, in the classical limit, the rapid rotation of the particle around the guiding centre and the slow guiding centre drift. In terms of these operators the Hamiltonian of the system can be rewritten as a power series in the magnetic length , and the fast and slow dynamics separates. The effective guiding centre Hamiltonian is obtained to second order in the adiabatic parameter and reproduces correctly the classical limit.
The motion of a neutral particle with a magnetic moment in an inhomogeneous magnetic field is considered. This situation, occurring, for example, in a Stern-Gerlach experiment, is investigated from classical and semiclassical points of view. It is assumed that the magnetic field is strong or slowly varying in space, i.e., that adiabatic conditions hold. To the classical model, a systematic Lie-transform perturbation technique is applied up to second order in the adiabatic-expansion parameter. The averaged classical Hamiltonian contains not only terms representing fictitious electric and magnetic fields but also an additional velocity-dependent potential. The Hamiltonian of the quantum-mechanical system is diagonalized by means of a systematic WKB analysis for coupled wave equations up to second order in the adiabaticity parameter, which is coupled to Planck's constant. An exact term-by-term correspondence with the averaged classical Hamiltonian is established, thus confirming the relevance of the additional velocity-dependent second-order contribution.
We study the valence electron of an alkaline atom or a general charged particle with arbitrary spin and with magnetic moment moving in a rotating magnetic field. By using a time-dependent unitary transformation, the Schr\"odinger equation with the time-dependent Hamiltonian can be reduced to a Schr\"odinger-like equation with a time-independent effective Hamiltonian. Eigenstates of the effective Hamiltonian correspond to cyclic solutions of the original Schr\"odinger equation. The nonadiabatic geometric phase of a cyclic solution can be expressed in terms of the expectation value of the component of the total angular momentum along the rotating axis, regardless of whether the solution is explicitly available. For the alkaline atomic electron and a strong magnetic field, the eigenvalue problem of the effective Hamiltonian is completely solved, and the geometric phase turns out to be a linear combination of two solid angles. For a weak magnetic field, the same problem is solved partly. For a general charged particle, the problem is solved approximately in a slowly rotating magnetic field, and the geometric phases are also calculated.
Abstract This review describes the results of investigations on charged particle dynamics in a magnetic field carried out over a number of years. The studies have unravelled the existence of some very surprising and unusual phenomena. Though existing on the macro-scale, they are found to be of quantum origin, and are thereby not covered by the Lorentz equation, which has been regarded conventionally as the descriptor of electrodynamic phenomena on the macro-scale. These novel phenomena have been found to be attributed to the ‘quantum modulation’ of the de Broglie wave along the magnetic field. This is brought about through the scattering-induced transition across Landau levels, leading to the modulation of the plane wave state along the field as a result of the entanglement between the parallel and perpendicular degrees of freedom. These findings were motivated by the predictions of a formalism developed by the author and include such unusual phenomena as (i) macro-scale matter wave interference effects and (ii) the detection of curl-free vector potential also on the macro-scale, both attributed to quantum modulation which is a matter wave on the macro-scale. The formalism is thus described as ‘macro-quantization of guiding centre motion’.
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.