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Sommerfeld Particle in Static Magnetic Field: Tunneling and Delayed Untwisting in Cyclotron

А. А. Власов

Published Apr 14, 2000• 0 citations
OpenAlex

Abstract

Motion of a charged particle with finite size, described by Sommerfeld model, in static magnetic field has two peculiar features: 1.) there is the effect of tunneling - Sommerfeld particle overcomes the barrier and finds itself in the forbidden, from classical point of view, area; 2.) the untwisting of trajectory in cyclotron for Sommerfeld particle is strongly delayed compared to that of a classical particle.

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Sommerfeld Particle in Static Magnetic Field: Tunneling and Delayed Untwisting in Cyclotron

А. А. Власов

Motion of a charged particle with finite size, described by Sommerfeld model, in static magnetic field has two peculiar features: 1.) there is the effect of tunneling - Sommerfeld particle overcomes the barrier and finds itself in the forbidden, from classical point of view, area; 2.) the untwisting of trajectory in cyclotron for Sommerfeld particle is strongly delayed compared to that of a classical particle.

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Motion of a charged particle with finite size, described by Sommerfeld model, in static magnetic field has two peculiar features: 1.) there is the effect of tunneling - Sommerfeld particle overcomes the barrier and finds itself in the forbidden, from classical point of view, area; 2.) the untwisting of trajectory in cyclotron for Sommerfeld particle is strongly delayed compared to that of a classical particle.
Quantum tunneling across a static potential barrier in a static magnetic field is very sensitive to an analytical form of the potential barrier. Depending on that, the oscillatory structure of the modulus of the wave function can be formed in the direction of tunneling. Due to an underbarrier interference, the probability of tunneling through a higher barrier can be larger than through a lower one. For some barriers the quantum interference of underbarrier cyclotron paths results in a strong enhancement of tunneling. This occurs in the vicinity of the certain magnetic field and is referred to as Euclidean resonance. This strongly contrasts to the Wentzel, Kramers, and Brillouin type tunneling which occurs with no magnetic field.
The motion of a particle with charge q and mass m in a magnetic field given by B = kB0 + B1 [icos(omegat) + jsin(omegat)] and an electric field which obeys Ñ ×E = - ¶ B/ ¶ t is analysed classically and quantum-mechanically. The use of a rotating coordinate system allows the analytical derivation of the particle classical trajectory and its laboratory wavefunction. The motion exhibits two resonances, one at omega = omegac = -qBo/m, the cyclotron frequency, and the other at omega = omegaL = -qBo/2m, the Larmor frequency. For omega at the first resonance frequency, the particle acquires a simple closed trajectory, and the effective hamiltonian can be interpreted as that of a particle in a static magnetic field. In the second case a term corresponding to an effective static electric field remains, and the particle orbit is an open line. The particle wave function and eigenenergies are calculated.
Application of the Bohr–Wilson–Sommerfeld quantization condition to a charged particle in a uniform magnetic field requires knowledge of the canonical momentum of such a particle, which in turn requires students to know about the vector potential. The canonical momentum in this situation is conventionally obtained by introducing an appropriate additional term involving the corresponding vector potential in the Lagrangian. In this work, we take a different approach to this problem by analyzing it with Maupertuis's principle of least action. We show that satisfying this principle for a charged particle moving in a uniform magnetic field requires that a term proportional to the flux passing through the area enclosed by the particle's trajectory be included in its action. This additional term accomplishes two tasks. First, it facilitates applying the Bohr–Wilson–Sommerfeld quantum condition for a charged particle moving in a uniform magnetic field without using the Lagrangian approach, and, second, it gives the appropriate canonical momentum and Lagrangian for a charged particle in a general electromagnetic field in a straightforward manner.