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Journal of Physics A Mathematical and General

Coherent state of an electron in a homogeneous constant magnetic field and the zero magnetic field limit

Sándor Varró

Published Jun 1, 1984• 28 citations
Publisher

Abstract

Coherent states of an electron embedded in a constant homogeneous magnetic field are constructed. The centres of the probability distributions belonging to these states gyrate along possible classical trajectories. Suitable packets of such coherent states are defined which reduce to properly normalised free electronic states in the zero magnetic field limit. A simple example is given to illustrate the dynamics of free electron localisation due to the presence of a magnetic field.

DOI
10.1088/0305-4470/17/8/019

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Coherent states of an electron embedded in a constant homogeneous magnetic field are constructed. The centres of the probability distributions belonging to these states gyrate along possible classical trajectories. Suitable packets of such coherent states are defined which reduce to properly normalised free electronic states in the zero magnetic field limit. A simple example is given to illustrate the dynamics of free electron localisation due to the presence of a magnetic field.
In the present work we construct coherent states in the magnetic–solenoid field, which is a superposition of the Aharonov–Bohm field and a collinear uniform magnetic field. In the problem under consideration there are two kinds of coherent states, those which correspond to classical trajectories which embrace the solenoid and those which do not. The constructed coherent states reproduce exactly classical trajectories, maintain their form under the time evolution and form a complete set of functions, which can be useful in semiclassical calculations. In the absence of the solenoid field these states are reduced to the well known in the case of uniform magnetic field Malkin–Man'ko coherent states.
We present the quantal dynamics of electronic wave packets in the hydrogen atom in the presence of circularly polarized microwave and magnetic fields in regions of classical stability. Whereas wave packets may disperse without the stabilizing influence of a magnetic field, in its presence stable motion can be maintained for wave packets that are localized either at global equilibria that are either a maximum or minimum in a zero-velocity surface. Because these extrema are locally harmonic their vacuum states are truly coherent states in the original sense of Schr\"odinger.
We deal with the motion of a charged particle in a constant magnetic field when immersed in a medium that exerts some type of friction. We analyse the problem classically and also quantum mechanically. In the latter case we use coherent state solutions of the problem that, for large energies, compared with the one associated with the cyclotron frequency, reduce to the classical limit. To study the dynamical behaviour we use time-dependent wave packets that can be constructed by superposition of the coherent states.
A complete set of coherent-state wave packets has been constructed for an electron in a uniform magnetic field. These states are nonspreading packets of minimum uncertainty that follow the classical motion. Use was made of the ladder operators that generate all the eigen-states of the Hamiltonian from any one energy eigenstate. The coherent states are the eigenstates of the two ladder operators that annihilate the zero-angular-momentum ground state. We have calculated the partition function, exploiting advantages of the coherent-state basis. The Landau diamagnetism and the de Haas-van Alphen oscillations are contained in the coherent-state framework.
The various quantum mechanical descriptions for an electron in a constant homogeneous magnetic field are considered for some particular gauges using the elementary methods of solutions. The internal relations between them are investigated by the quantum mechanical operator technique to get the systematic solutions for the wave functions in coordinate space. Then the transformation theory between various solutions is established and further the oscillating motion of a wave packet is analyzed to present another direct example of tbe theorem due to Ehrenfest.