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Journal of Physics A Mathematical and General

Evolution of moments over quantum wavepackets or classical clusters

M Andrews, Michael J. W. Hall

Published Jan 11, 1985• 8 citations
Publisher

Abstract

Using a symplectic notation which allows the equations to apply equally to a wavepacket representing a spin-less quantum particle or to a cluster of identical classical particles without mutual interactions, the study of the time-evolution of moments is extended to arbitrary orders for arbitrary Hamiltonians. Appropriately symmetrised moments for the quantum case are seen to play a special role and correspond very closely to the classical moments.

DOI
10.1088/0305-4470/18/1/014

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Using a symplectic notation which allows the equations to apply equally to a wavepacket representing a spin-less quantum particle or to a cluster of identical classical particles without mutual interactions, the study of the time-evolution of moments is extended to arbitrary orders for arbitrary Hamiltonians. Appropriately symmetrised moments for the quantum case are seen to play a special role and correspond very closely to the classical moments.
For a wavepacket representing a particle subject to a conservative force consideration is given to the evolution in time of the mean position and momentum and to the evolution of the spread of these quantities as measured by the mean square deviation from the mean. A closed system of equations involving these spreads is obtained by expanding the potential in powers about the mean position and by neglecting terms of third order or higher in the deviations from the mean, an approximation appropriate when the force does not vary too much over the width of the packet. These equations are solved in terms of the trajectories of a classical time-dependent oscillator. These trajectories can be found by differentiation of the trajectories for the force under consideration. In more than one dimension, or for more than one particle, the appropriate generalisation of the spread is the set of second-order correlations.
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.
The classical Hamiltonian of a point charged particle with intrinsic spin, which is composed of the relativistic orbit Hamiltonian and the spin Hamiltonian. This spin Hamiltonian can be derived from the Thomas-Bargmann-Michel-Telegdi equation. In our thesis, we will prove that the classical Hamiltonian in which the gyromagnetic ratio equals 2 is in agreement with the low-energy limit of the Dirac equation, the relativistic quantum theory for a spin-1/2 charged particle. For this purpose, we apply the Foldy-Wouthuysen transformation to the Dirac Hamiltonian under the assumptions of the static and homogeneous electromagnetic fields. Furthermore, in order to maintain such a correspondence in arbitrary gyromagnetic ratio cases, we perform the F-W transformation to the Dirac-Pauli Hamiltonian, which is the Dirac Hamiltonian plus the anomalous magnetic dipole moment Hamiltonian. Up to the twelfth-order of F-W transformed Dirac and Dirac-Pauli Hamiltonian, the calculation results are in accord with the corresponding classical Hamiltonians.
Abstract Emergence of classicality from quantum mechanics, a hotly debated topic, has had no satisfactory resolution so far. Various approaches including decoherence and gravitational interactions have been suggested. In the present work, the Schrödinger–Newton model is used to study the role of semi‐classical self‐gravity in the evolution of massive spin‐1/2 particles in a Stern‐Gerlach experiment. For small mass, evolution of the initial wavepacket in a spin superposition shows a splitting in the magnetic field gradient into two trajectories as in the standard Stern–Gerlach experiment. For larger mass, the deviations from the central path are less than in the standard Stern–Gerlach case, while for high enough mass, the wavepacket does not split, and instead follows the classical trajectory for a magnetic moment in inhomogeneous magnetic field. This indicates the emergence of classicality due to self‐gravitational interaction when the mass is increased. In contrast, decoherence which is a strong contender for emergence of classicality, leads to a mixed state of two trajectories corresponding to the spin‐up and spin‐down states, and not the classically expected path. The classically expected path of the particle probably cannot be explained even in the many‐worlds interpretation of quantum mechanics. Stern–Gerlach experiments in the macroscopic domain are needed to settle this question.