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European Journal of Physics

Quantum systems that follow classical dynamics

Giovanni Manfredi, S Mola, Marc Roy Feix

Published May 1, 1993• 14 citations
Publisher

Abstract

For a special class of potentials, the dynamical evolution of any quantum wavepacket is entirely determined by the laws of classical mechanics. Here, the properties of this class are investigated both from the viewpoint of the Ehrenfest theorem (which provides the evolution of the average position and momentum), and the Wigner representation (which expresses quantum mechanics in a phase space formalism). Finally, these results are extended to the case of a charged particle in a uniform magnetic field.

DOI
10.1088/0143-0807/14/3/002

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For a special class of potentials, the dynamical evolution of any quantum wavepacket is entirely determined by the laws of classical mechanics. Here, the properties of this class are investigated both from the viewpoint of the Ehrenfest theorem (which provides the evolution of the average position and momentum), and the Wigner representation (which expresses quantum mechanics in a phase space formalism). Finally, these results are extended to the case of a charged particle in a uniform magnetic field.
A semiclassical approximation is derived by using a family of wavepackets to map arbitrary wavefunctions into phase space. If the Hamiltonian can be approximated as linear over each individual wavepacket, as often done when presenting Ehrenfest's theorem, the resulting approximation is a linear first-order partial differential equation on phase space, which will be referred to as the Schrödinger-Ehrenfest or SE equation. This advectively transports wavefunctions along classical trajectories, so that as a trajectory is followed in time the amplitude remains constant while the phase changes by the action divided by ℏ\hbarℏ. The wavefunction's squared-magnitude is a plausible phase space density and obeys Liouville's equation for the classical time evolution. This is a derivation of the Koopman-von~Neumann (KvN) formulation of classical mechanics, which previously was postulated but not derived. With the time-independent SE equation, continuity of the wavefunction requires the change of phase around any closed path in the torus covered by a classical trajectory to be an integer multiple of 2π2π2π, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.
Abstract Ehrenfest theorem is extended to the relativistic quantum theory of charged particles, moving under the influence of a classical electromagnetic field. In order to extend the original Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton’s second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the conserved electric four-current built from the particle wave function. The derivation is based on the conservation laws for the energy and momentum.
Abstract Ehrenfest's theorem for a system of electrons in a time‐dependent external force is the quantal analogue to Newton's second law of motion for interacting classical particles. The theorem is in terms of the averaged force and position of the electrons. Here we draw the quantal analogue to the equation of motion for the individual classical particle and thereby describe the internal forces experienced by each electron and the evolution of these forces with time. This explanation is based on the description of Schrödinger theory in terms of fields and their quantal sources. The quantum mechanical average of the internal fields and their averaged torques, taken over all the electrons, vanish as they must. However, unlike classical physics, the vanishing of the averaged internal fields cannot be attributed solely to Newton's third law but is additionally a consequence of quantum mechanics. The field perspective thus leads to a new derivation of Ehrenfest's theorem. Furthermore, the theorem can be expressed in terms of the response of the electrons to the external field as described by a field representative of the electronic current density. In addition, the vanishing of the averaged torque of the internal fields leads to a quantal torque–angular momentum equation analogous to that of classical physics. The structure of the internal fields is demonstrated for both a ground and an excited state of the analytically solvable Hooke's atom. The concept of the internal field is extended via quantal density functional theory to the S system of noninteracting Fermions with density equivalent to that of the Schrödinger theory of electrons. By invoking Ehrenfest's theorem and the quantal torque–angular momentum relationship, sum rules for the S system are derived. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
We consider a charged particle driven by a time-dependent flux threading a quantum ring. The dynamics of the charged particle is investigated using a classical treatment, a Fourier expansion technique, a time-evolution method, and the Lewis-Riesenfeld approach. We have shown that, by properly managing the boundary conditions, a time-dependent wavefunction can be obtained using a general non-Hermitian time-dependent invariant, which is a specific linear combination of initial angular-momentum and azimuthal-angle operators. It is shown that the linear invariant eigenfunction can be realized as a Gaussian-type wavepacket with a peak moving along the classical angular trajectory, while the distribution of the wavepacket is determined by the ratio of the coefficient of the initial angle to that of the initial canonical angular momentum. From the topologically nontrivial nature as well as the classical trajectory and angular momentum, one can determine the dynamical motion of the wavepacket. It should be noted that the peak position is no longer an expectation value of the angle operator, and hence the Ehrenfest theorem is not directly applicable in such a topologically nontrivial system.