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American Journal of Physics

A wave-packet description of the motion of a charged particle in a uniform magnetic field

William Zimmermann

Published Jul 1, 1989• 3 citations
Publisher

Abstract

A discussion is given of the construction of wave packets to describe the motion of a charged particle in a uniform magnetic field. Examples are given of the construction of wave packets from eigenstates of the Hamiltonian for both the Landau gauge in Cartesian coordinates and the cylindrically symmetric gauge in cylindrical coordinates.
DOI
10.1119/1.15952

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Comment on ‘‘A wave-packet description of the motion of a charged particle in a uniform magnetic field,’’ by W. Zimmermann, Jr. [Am. J. Phys. 5 7, 593–598 (1989)]

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The eigensolutions of the Schrödinger equation for the motion of a charged particle in a uniform magnetic field depend on the choice of electromagnetic gauge. In the Landau gauge it is hard to discern a relationship between the eigensolutions and the circular or helical orbits of classical theory. In this paper a wave packet is constructed from the Landau gauge eigensolutions which simulates the classical motion in the plane perpendicular to the magnetic field. Some properties of the wave packet are discussed.
In the present work we treat the problem of a particle in a uniform magnetic field along the symmetric gauge, so chosen since the wavefunctions present the required cylindrical symmetry. It is our understanding that by means of this work we can make a contribution to the teaching of the present subject, as well as encourage students to use computer algebra systems in solving problems of quantum mechanics. We obtained the degeneracy of the spectrum of eigenvalues in a very clear way. Through the use of a computer algebra system we show graphs of the probability density associated with different eigenvalues as well as compare such functions for some degenerate states, which helps us to visualize the physics of the problem. We also present a semiclassical model which gives a physical insight regarding the paradoxical fact that eigenfunctions associated with opposite angular momenta and different energy eigenvalues have the same probability density. Finally, by solving the time-dependent Schrodinger equation we obtain the time evolution of a wave packet that at time zero was considered to be localized in a definite region of the lattice. The centroid of such a packet performs an orbit similar to that obtained in the classical treatment of a particle in a magnetic field.
Double wave quantum theory is applied to describe the motion of charged particle in a uniform magnetic field. This description of the motion of the particle is complete. One can tell the value of an arbitrary physical quantity at any time. The probability and the formula to calculate mean values in ordinary quantum mechanics comes from the average of double wave description over a certain ensemble. The rule of gauge transformation in quantum mechanics can be seen from the average over an ensemble.