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College Physics

Quantum-classical correspondence for a charge in a uniform magnetic field

Quanhui Liu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The motion of a charged particle in a uniform magnetic field forms a helical curve.In the plane perpendicular to the magnetic field,the orbit is a circular.In quantum mechanics,the operator form of the classical circle relations by gauge transformation is derived.

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In the quantum field,due to Bohr correspondence principle,quantum mechanics go to classical mechanics in large quantum number.Based upon Heisenberg correspondence principle,quantum matrix element of a Hermitian operator reduces to the coefficient of Fourier expansion of the corresponding classical quantity in the classical limit.Using Heisenberg correspondence principle,quantum-classical correspondence of particles in magnetic is studied.Applying Heisenberg correspondence principle in relativistic realm,the exact wave functions of Klein-Gordon equation are obtained for particles in magnetic with quantum matrix element in a rectangular coordinate system of new representation.It turns out that in the classical limit matrix elements of quantum operators reduces to the classical solutions and with classical approach,the coordinate is cyclical variation with time,the orbit of particles is a circle,and the movement is uniform circular motion in uniform magnetic field.
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 Newtonian mechanics,the motion of the charge in magnetic field is in equidistance helical line.It is composed of the uniform motion in magnetic field and the circular motion vertical to the magnetic field.In quantum mechanics,its motion is two-dimensional resonance.Its energy level is separate.Its wave function is probability.With this theory to explain magnetic focusing a large number of charge moving in the magnetic field are not at a point 100%.Instead,they wwill be dispersed according to probability.
By using appropriate gauge transformations one can develop quasiclassical wavefunctions for charged particles in a uniform magnetic field which have large amplitudes only in the neighborhood of classical Larmor orbits centered at arbitrary points in a plane perpendicular to the field. Pairs of such wavefunctions are not orthogonal to each other. Interesting geometrical relationships between these solutions and the degenerate orthonormal sets of wave-functions in a particular gauge can be found as a result of evaluating overlap integrals. This has been done for both the cylindrical and Landau gauges. In the cylindrical gauge certain of the orthonormal wavefunctions corresponding to different energies give identical probability distributions. Relationships between quantum and classical results, and the effect of the uncertainty principle are discussed.
Using the Landau and symmetric gauges for the vector potential of a constant magnetic field and the quantum problem of a charged particle moving on a flat surface, we show the classical electromagnetic gauge transformation does not correspond to a one-dimensional unitary group transformation U(1) of the wave function for the quantum case. In addition, with the re-examination of the relation between the magnetic field B and its vector potential A, we found that, in order to have a consistent formulation of the dynamics of the charged particle with both expressions, we must have that B=∇×A if and only if B≠0.