In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [ Phys. Rev. X , 4 , 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [ Phys. Rev. X , 4 , 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.

arXiv:1605.01714Journal of Physics Conference SeriesMar 1, 201610 citations

Classical and semi-classical limits in phase space

Clay D. Spence

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 $\hbar$ . 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 $2π$ , giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.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 $\hbar$. 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 $2π$, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.

arXiv:2305.18644May 29, 20230 citations

Quantum Dynamics with Trajectories

Springer eBooksJan 1, 2005141 citations

Applicability criterion for semiclassical Bohmian dynamics

Sophya V. Garashchuk, Vitaly A. Rassolov

Abstract In this paper we develop a criterion of the applicability for the trajectory‐based semiclassical propagation methods that are exact for Gaussian wavepackets in locally quadratic potentials. The error estimate takes into account a deviation of the wavefunction from the Gaussian form due to the anharmonicity of a potential. The derivation is based on the time‐dependent noncoherent solutions of the Schrödinger equation for a general quadratic potential that form a complete orthonormal time‐dependent basis. The anharmonicity is treated as a time‐dependent cubic perturbation, which is further related to the nonlinearity of a force acting on trajectories. The criterion is expressed in terms of scalar products of the average force, its first moments, and dispersion of the wavefunction and is efficient in the context of dynamics with the Bohmian trajectories. We give an application for the Bohmian dynamics with the approximate quantum potential, for the Gaussian wavepacket dynamics and for the semiclassical initial value representation propagator. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004Abstract In this paper we develop a criterion of the applicability for the trajectory‐based semiclassical propagation methods that are exact for Gaussian wavepackets in locally quadratic potentials. The error estimate takes into account a deviation of the wavefunction from the Gaussian form due to the anharmonicity of a potential. The derivation is based on the time‐dependent noncoherent solutions of the Schrödinger equation for a general quadratic potential that form a complete orthonormal time‐dependent basis. The anharmonicity is treated as a time‐dependent cubic perturbation, which is further related to the nonlinearity of a force acting on trajectories. The criterion is expressed in terms of scalar products of the average force, its first moments, and dispersion of the wavefunction and is efficient in the context of dynamics with the Bohmian trajectories. We give an application for the Bohmian dynamics with the approximate quantum potential, for the Gaussian wavepacket dynamics and for the semiclassical initial value representation propagator. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum ChemistryAug 30, 20042 citations

WAVEPACKET APPROACHES TO DISSIPATIVE QUANTUM DYNAMICS

Fausto Martelli

Dec 14, 20100 citations

\hbar

. 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

2π

, giving a standing wave picture of old quantum mechanics. While this applies to any system, for separable systems it gives Bohr-Sommerfeld quantization.