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European Journal of Physics

A charged particle on a ring in a magnetic field: quantum revivals

G. A. Vugalter, A. K. Das, V. A. Sorokin

Published Nov 28, 2003• 9 citations
Publisher

Abstract

This paper deals with a simple quantum system which, in addition to being of general interest, can be discussed in an advanced undergraduate course on quantum mechanics. The system consists of a charged particle on a ring of fixed radius—a rigid charged rotator. For this system we have analytically investigated the influence of a dc magnetic field and the symmetry of an initial wavefunction on the scenario of perfect quantum revivals. Using Gisin’s model, we discuss the decay of revivals in the presence of dissipation. Some of our analytical results have been numerically illustrated for Gaussian coherent states. We have also studied the influence of a δ-potential on quantum revivals in this system.

DOI
10.1088/0143-0807/25/2/003

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This paper deals with a simple quantum system which, in addition to being of general interest, can be discussed in an advanced undergraduate course on quantum mechanics. The system consists of a charged particle on a ring of fixed radius—a rigid charged rotator. For this system we have analytically investigated the influence of a dc magnetic field and the symmetry of an initial wavefunction on the scenario of perfect quantum revivals. Using Gisin’s model, we discuss the decay of revivals in the presence of dissipation. Some of our analytical results have been numerically illustrated for Gaussian coherent states. We have also studied the influence of a δ-potential on quantum revivals in this system.
Quantum mechanical objects or nano-objects have been proposed as bits for information storage. While time-averaged properties of magnetic, quantum-mechanical particles have been extensively studied experimentally and theoretically, experimental investigations of the real time evolution of magnetization in the quantum regime were not possible until recent developments in pump–probe techniques. Here we investigate the quantum dynamics of effective spin systems by means of analytical and numerical treatments. Particular attention is paid to the quantum revival time and its relation to the magnetization tunneling. The quantum revival time has been initially defined as the recurrence time of a total wave-function. Here we show that the quantum revivals of wave-functions and expectation values in spin systems may be quite different which gives rise to a more sophisticated definition of the quantum revival within the realm of experimental research. Particularly, the revival times for integer spins coincide which is not the case for half-integer spins. Furthermore, the quantum revival is found to be shortest for integer ratios between the on-site anisotropy and an external magnetic field paving the way to novel methods of anisotropy measurements. We show that the quantum tunneling of magnetization at avoided level crossing is coherent to the quantum revival time of expectation values, leading to a connection between these two fundamental properties of quantum mechanical spins.
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.
We study the motion of a quantum charged particle, constrained on the surface of a cylinder, in the presence of a radial magnetic field. When the spin of the particle is neglected, the system essentially reduces to an infinite family of simple harmonic oscillators, equally spaced along the axis of the cylinder. Interestingly enough, it can be used as a quantum Fourier transformer, with convenient visual output. When the spin-1/2 of the particle is taken into account, a non-conventional perturbative analysis results in a recursive closed form for the corrections to the energy and the wavefunction, for all eigenstates, to all orders in the magnetic moment of the particle. A simple two-state system is also presented, the time evolution of which involves an approximate precession of the spin perpendicularly to the magnetic field. A number of plots highlight the findings while several three-dimensional animations have been made available on the web.
On a ring threaded by a flux line moves a charged quantum particle whose rotation is hindered by an angle-dependent potential. When the potential is rigidly rotated through , the particle (in the n th eigenstate of the potential on the ring) acquires a geometric phase . A general formula is , where is the dimensionless quantum flux and is a Schrödinger current associated with the state. Properties of are obtained in terms of the transmission coefficient round the ring. vanishes when the box is impenetrable or when is integer or half-integer. Energy levels form bands, with playing the role of Bloch pseudomomentum. For unhindered semiclassical states above the barrier, a WKB theory gives the geometric phase and hence the (previously calculated) classical Hannay angle . This does not vanish when is integer or half-integer, but these cases correspond to band edges where the semiclassical states are degenerate and differ greatly from the true asymptotic states. The theory is illustrated by the exact calculation of for a model where the potential is a delta function.
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.
We analyzed spin polarization dynamics in a two-dimensional system of spin 1/2 charged particles with spin-orbit interaction in perpendicular magnetic field in the presence of external noise. It was shown that spin polarization reveals quantum oscillations, collapses, and revivals. The hierarchy of time scales corresponding to quantum oscillations, collapses, and revivals was identified and analyzed. It was demonstrated that variation of external magnetic field and noise intensity results in the significant changing of spin fluctuations spectra, which modifies from the triplet structure to the well resolved fine structure.