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arXiv:1111.6107

Electron wave-functions in a magnetic field

D. K. Sunko

Published Nov 25, 2011• 0 citations
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Abstract

The problem of a single electron in a magnetic field is revisited from first principles. It is shown that the standard quantization, used by Landau, is inconsistent for this problem, whence Landau's wave functions spontaneously break the gauge symmetry of translations in the plane. Because of this Landau's (and Fock's) wave functions have a spurious second quantum number. The one-body wave function of the physical orbit, with only one quantum number, is derived, and expressed as a superposition of Landau's wave functions. Conversely, it is shown that Landau's wave functions are a limiting case of physical solutions of a different problem, where two quantum numbers naturally appear. When the translation gauge symmetry is respected, the degeneracy related to the choice of orbit center does not appear in the one-body problem.

DOI
10.48550/arxiv.1111.6107

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The problem of a single electron in a magnetic field is revisited from first principles. It is shown that the standard quantization, used by Landau, is inconsistent for this problem, whence Landau's wave functions spontaneously break the gauge symmetry of translations in the plane. Because of this Landau's (and Fock's) wave functions have a spurious second quantum number. The one-body wave function of the physical orbit, with only one quantum number, is derived, and expressed as a superposition of Landau's wave functions. Conversely, it is shown that Landau's wave functions are a limiting case of physical solutions of a different problem, where two quantum numbers naturally appear. When the translation gauge symmetry is respected, the degeneracy related to the choice of orbit center does not appear in the one-body problem.
We summarize our results about the quantization of a charged particle motion without spin inside a flat box under a static electromagnetic field with Landau’s gauge for the magnetic field, where Fourier’s transformation was used to analyze the problem, to point out that there exists a wave function which is different to that one given by Landau with the same Landau’s levels. The quantization of the magnetic flux is deduced differently to previous one, and a new solution is presented for the case of symmetric gauge of the magnetic field, and having the same Landau’s levels.
The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).
Abstract The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).
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.
Expansions of a plane wave in Landau states are derived corresponding to an adiabatic and to a sudden switching on of the magnetic field. Two earlier solutions given by Faisal (1982) and by Ohsaki (1983) are found to be valid alternative expansions in the adiabatic case, which differ only in the choice of normalisations of the basis states used. A result identical to Ohsaki's is reobtained, since the Landau states are confined by conservation of energy to a large but finite area L x L y approximately 1/ gamma , perpendicular to the field. This energy conservation requires the corresponding sum over the degenerate Landau states to be restricted within a finite range. The upper and lower limits of this range are derived. The completeness relation of Faisal's expansion shows that it is also effectively and automatically confined to the same range of states. In the limit gamma to 0 both sets of expansion coefficients become identical and the results go over to the desired plane-wave state. In the sudden case, when the field switches instantaneously from zero to a finite value, the expansion no longer consists of Landau states of a fixed energy, and the result is new. The two results correspond to different limiting cases of any experimental situation, and the sudden expansion is shown to be invalid at the highest laboratory field ( approximately 20 kG) in current use. The experimental situations corresponding to the two different expansions are discussed.
The Schr\"odinger equation for an interacting spinless electron gas in a nonuniform magnetic field admits an exact solution in Jastrow product form when the fluctuations in the magnetic field track the fluctuations in the scalar potential. For tracking realizations in a two-dimensional electron gas, the degeneracy of the lowest Landau level persists and the ``tracking'' solutions span the ground state subspace. In the context of the fractional quantum Hall problem, the Laughlin wave function is shown to be a tracking solution. Tracking solutions for screened Coulomb interactions are also constructed. The resulting wave functions are proposed as variational wave functions with potentially lower energy in the case of non-negligible Landau level mixing than the Laughlin function.