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American Journal of Physics

Charged Particle in a Low Magnetic Field with Finite Boundary Conditions

R.D. Zwicker, Frank A. Butler

Published Sep 1, 1973• 0 citations
Publisher

Abstract

Exact wave functions are given for a charged particle in a uniform magnetic field, confined in one direction between two infinite potential barriers. The manner in which these wave functions proceed in the zero-field limit to free particle plane waves is explored with the aid of the WKB (Wentzel-Kramers-Brillouin) formalism. The energy eigenspectrum in the low-field case is found as the zero-field eigenspectrum plus correction terms in powers of the magnetic field. The form of the correction terms and other characteristics of the quantum mechanical problem are explained by comparison with the analogous problem in classical mechanics.

DOI
10.1119/1.1987480

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Exact wave functions are given for a charged particle in a uniform magnetic field, confined in one direction between two infinite potential barriers. The manner in which these wave functions proceed in the zero-field limit to free particle plane waves is explored with the aid of the WKB (Wentzel-Kramers-Brillouin) formalism. The energy eigenspectrum in the low-field case is found as the zero-field eigenspectrum plus correction terms in powers of the magnetic field. The form of the correction terms and other characteristics of the quantum mechanical problem are explained by comparison with the analogous problem in classical mechanics.
In studying the diamagnetism of free electrons in a uniform magnetic field it was found that reducing the field to zero in the wavefunction did not yield the experimentally indicated free particle plane wave wavefunction. However, solving the Schroedinger Equation resulting from setting the field equal to zero in the original equation did yield a plane wave wavefunction. This paradox was not found to be peculiar to the case of a charged particle in a uniform magnetic field but was found to occur in a number of other systems. In order to gain an understanding of this unexpected behavior, the following systems were analyzed: the one-dimensional square well potential; a charged, spinless particle in a Coulomb field and in a uniform electric field; a one-dimensional harmonic oscillator; and a charged, spinless particle in a uniform magnetic field. From these studies the following were obtained: conditions for determining the result of reducing the potential in a wavefunction; the condition under which the potential of a system may be switched off while maintaining the energy of the system constant; the relationship between the result of physically switching off a potential, the result of reducing it in the wavefunction, and the solution of the Schroedinger Equation obtained by decreasing the potential to zero in the original wave equation; and a general property of any wavefunction with respect to reducing any parameter within this wavefunction.
We make three remarks about the quantum theory of a charged particle moving in a time-independent, uni-directional magnetic field (which we take to be parallel to the z-axis). First, when the field is uniform, we comment on the relation between the wave-function on the one hand if it is chosen to be an eigenstate of momentum (leaving aside the trivial z-dependence) and hence non-normalizable, and on the other hand if it is chosen to be an eigenstate of angular momentum. Second, we discuss the form of the wave-function when the magnetic field had a weak z-dependence. Finally, we make a short comment on the derivation of the diamagnetism of a free electron gas.
One-dimensional particle states are constructed according to orthogonality conditions, without requiring boundary conditions. Free particle states are constructed using Dirac's delta function orthogonality conditions. The states (doublets) depend on two quantum numbers: energy and parity. With the aid of projection operators the particles are confined to a constrained region, in a way similar to the action of an infinite well potential. From the resulting overcomplete basis only the mutually orthogonal states are selected. Four solutions are found, corresponding to different non-commuting Hamiltonians. Their energy eigenstates are labeled with the main quantum number n and parity "+" or "-". The energy eigenvalues are functions of n only. The four cases correspond to different boundary conditions: (I) the wave function vanishes on the boundary, (II) the derivative of the wavefunction vanishes on the boundary,(III) periodic (symmetric) boundary conditions, (IV) periodic (antisymmetric)boundary conditions . Among the four cases, only solution (III) forms a complete basis in the sense that any function in the constrained region, can be expanded with it. By extending the boundaries of the constrained region to infinity, only solution (III) converges uniformly to the free particle states. Orthogonality seems to be a more basic requirement than boundary conditions. By using projection operators, confinement of the particle to a definite region can be achieved in a conceptually simple and unambiguous way, and physical operators can be written so that they act only in the confined region.