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Physical review. B, Solid state

Polarization effects in penetration of a barrier with a magnetic field applied

J. Schmit, Robert H. Good

Published Aug 1, 1977• 4 citations
Publisher

Abstract

The spin polarization of an electron moving nonrelativistically through a potential barrier $V(z)$, with also a constant uniform magnetic field ($0,0,B$) applied, is calculated using the WKB approximation for the $z$ dependence of the wave function. It is found that, in a classically allowed region, the polarization precesses at fixed angle $\ensuremath{\theta}$ about the magnetic field, with phase angle the same as a classical particle with a magnetic dipole moment. The particle penetrates through the barrier with no change in the phase of the precession but with a discontinuity in $\ensuremath{\theta}$.

DOI
10.1103/physrevb.16.1197

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The spin polarization of an electron moving nonrelativistically through a potential barrier V(z)V(z)V(z), with also a constant uniform magnetic field (0,0,B0,0,B0,0,B) applied, is calculated using the WKB approximation for the zzz dependence of the wave function. It is found that, in a classically allowed region, the polarization precesses at fixed angle \ensuremathθ\ensuremath{\theta}\ensuremathθ about the magnetic field, with phase angle the same as a classical particle with a magnetic dipole moment. The particle penetrates through the barrier with no change in the phase of the precession but with a discontinuity in \ensuremathθ\ensuremath{\theta}\ensuremathθ.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
The present paper shows that the nature of the polarization of charged spin-1/2 particles moving in a uniform magnetic field changes dramatically in a relatively weak transverse axisymmetric magnetic field. The direction along which the spin projection is quantized has a fixed orientation with respect to the axes of a cylindrical coordinate system and can form a substantial angle with the direction of the uniform magnetic field. The presence of spin quantization is proved both by the fact that the commutator of the Hamiltonian operator and the projection of the polarization operator in the direction of quantization is zero and by analyzing the Bargmann-Michel-Telegdi equation for this given case. Finally, the possibilities of detecting this effect and utilizing it are discussed.
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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.
If a spin-½ particle is placed in a constant uniform magnetic field for a time t during which, classically, the Larmor precession angle would be θ, then according to quantum mechanics the components of the spin part of the wave function undergo phase changes of ±θ/2. Hence, corresponding to a classical precession of 2π radians, there is a sign change of the spinor, whatever the initial polarization may be. Bernstein pointed out that this sign change can be observed by coherently splitting a fermion beam, subjecting one part to the magnetic field, and coherently recombining the parts. Evidently, the same interference effect can be achieved by subjecting the two parts of the beam to fields which would classically cause precessions of π radians in opposite senses. He noted advantages in studying neutral rather than charged particles, and outlined a proposal for an interferometry experiment with neutrons. The recent invention and application of a remarkable single-crystal neutron interferometer greatly increases the feasibility of observing the sign change. In the experiment of Ref. 3 a neutron beam is split, and the two parts follow paths of different elevations. The interference of the recombined parts exhibits the effect of gravitational potential difference between the two elevations. In the experiment which we propose, the paths could be kept at the same elevation, but phase differences would be produced by appropriate magnetic fields.