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Journal of Physics A Mathematical and General

Coherent states and dissipation for the motion of a charged particle in a constant magnetic field

Dieter Schuch, M. Moshińsky

Published May 28, 2003• 23 citations
Publisher

Abstract

We deal with the motion of a charged particle in a constant magnetic field when immersed in a medium that exerts some type of friction. We analyse the problem classically and also quantum mechanically. In the latter case we use coherent state solutions of the problem that, for large energies, compared with the one associated with the cyclotron frequency, reduce to the classical limit. To study the dynamical behaviour we use time-dependent wave packets that can be constructed by superposition of the coherent states.

DOI
10.1088/0305-4470/36/23/320

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We deal with the motion of a charged particle in a constant magnetic field when immersed in a medium that exerts some type of friction. We analyse the problem classically and also quantum mechanically. In the latter case we use coherent state solutions of the problem that, for large energies, compared with the one associated with the cyclotron frequency, reduce to the classical limit. To study the dynamical behaviour we use time-dependent wave packets that can be constructed by superposition of the coherent states.
Abstract Based on the quasi-classical approximation, a general approach is proposed for constructing non-stationary quantum states of a charged particle in a magnetic field, when the dissipative forces of viscous friction and drag, proportional to the velocity and the square of the velocity, respectively, are also significant. The corresponding quasi-classical Green’s function is found, with the help of which the squeezed and coherent states of the particle are studied. It is shown that the dissipation and a magnetic field suppress the quantum properties of the particle. This is especially true for the transverse motion with respect to the magnetic field. Over time, the coherent and squeezed states transform into the same static state, which is characterized by a zero uncertainty of the transverse coordinates and an uncertainty of the longitudinal coordinate, which contains information about the initial velocity of the particle.
Abstract A quasi-classical approach is proposed for describing the motion of a charged particle in a magnetic field, taking into account irreversible losses due to the macroscopic viscosity of the medium. The wave function of a charged particle corresponding to its quasi-classical coherent state is found. It is shown that viscosity leads to an irreversible collapse of the wave function in directions perpendicular to the magnetic field. At the same time along the magnetic field the wave function experiences irreversible spreading up to a certain static limit. Thus, in the transverse directions a viscous medium and a magnetic field behave like a classical measuring device. In the longitudinal directions the signs of the quantum Zeno effect are visible. As a result of such anisotropic quantum dynamics the wave packet of the probability density takes the form of a thin filament wound around a magnetic field. The length of the filament is determined by the limiting value of the uncertainty of the longitudinal coordinate of the particle. In turn, this asymptotic uncertainty contains the information about its initial value, about the mass of the particle, and about the properties of the viscous medium.
It has been shown that the irreversible dynamics of a system in a dissipative environment can be described by an effective one-particle Hamiltonian (in the classical case) or an effective one-particle Schrödinger equation (in the quantum mechanical case) without taking into account the environmental degrees of freedom explicitly. In the quantum mechanical case, the description can be achieved either by a nonlinear modification of the Schrödinger equation or an explicitly time-dependent Hamiltonian operator — where both forms are connected via a non-unitary transformation, corresponding to a non-canonical transformation on the classical level. Taking into account an external magnetic field, in both formalisms, an additional unphysical gauge-dependent term occurs in the equations of motion. This shortcoming can be eliminated if the Lagrangian and Hamiltonian that provide Maxwell’s equations of electrodynamics undergo, in the dissipative case, the same kind of transformation as the mechanical degrees of freedom of the system. As a result, a consistent description in terms of exact analytical wave packet-solutions for the dissipative motion in a magnetic field can be obtained that will be discussed in detail.
A system of two charged particles in a harmonic trap with additional magnetic field is considered. The problem is reduced to a single-particle one in relative coordinates. The ground- and lowest excited-state energies and wave functions are found. The ground state exhibits non-zero expectation value of the velocity (kinetic momentum) and the probability current density does not vanish as well. When the ground state becomes degenerate the expectation value of velocity becomes discontinuous. The effects associated with turning on of the magnetic field are studied by solving the appropriate time-dependent Schrödinger equation. No substantial differences between abrupt (discontinuous in time) and continuous switching on have been observed. Evolution of a wave packet which is initially Gaussian is also investigated. The wave packet loses its Gaussian nature and, after sufficiently large time, a system of diffractive maxima and minima is built.
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.
Abstract In this paper we show how the classical result of a charged particle moving in a circle in the xy plane, when a uniform magnetic field is directed along the z -axis, can be derived from the Landau quantum theory using the coherent states of the two-dimensional isotropic harmonic oscillator in the xy plane. The coherent states in this case are the simultaneous eigen vectors of the annihilation operators a + and a − . We prove that the time-dependent coordinate space wave packets representing the time-dependent coherent states move in a circle with the cyclotron frequency ω c = ∣ q ∣ B m and with a radius given by the classical expression, but given in terms of the quantum mechanical expectation values. The expectation value of the energy of the particle and of the square of the radius of its circular are proportional to the square of the magnitude of the eigen value of a + in the coherent state, where as the x and y coordinates of the centre of the circle are proportional to the real and the imaginary parts of the eigen value of a − . The phase of the circular motion is the same as the phase of the complex eigen value of a + . So for a given energy of the particle or for a given radius of the circular orbit, there are an infinite number of circles which differ from each other by the x and y coordinates of the centre as well as the phase of the circular motion. The infinite degeneracy of the Landau levels is due to the invariance of the energy eigen values under spatial translations in the xy plane and rotations about the z -axis. We also show that as the magnitude of the eigen value of a + becomes much larger than one, the relative uncertainty or fluctuation in the energy and in the radius of the circular orbit becomes negligibly small as we expect for a classical state.