Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Theoretical Physics

Velocity Operators and Time-Energy Relations in Relativistic Quantum Mechanics

Mario Bunge

Published Jan 1, 2003• 19 citations
Publisher
DOI
10.1023/a:1023395525747

Related papers

Time and energy operators in quantum mechanics

Chun-Fang Li, Qi Wang

Journal of Shanghai University (English Edition)•Mar 1, 1998•1 citation

A dynamical time operator in Dirac's relativistic quantum mechanics

M. Bauer

A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.

arXiv:0908.2789•International Journal of Modern Physics A•Mar 4, 2014•38 citations

Extension of a space-time formalism in relativistic quantum mechanics

C. A. Lopez, M. A. Pérez

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento•Feb 1, 1981•4 citations

A dynamical time operator in relativistic quantum mechanics

M. Bauer

A self-adjoint dynamical time operator is introduced in Dirac’s relativistic formulation of quantum mechanics and shown to satisfy with the Hamiltonian a commutation relation analogous to that of the position and momentum operators. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie’s hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli’s objection is shown to be resolved or circumvented. Possible relevance to current developments in interference in time and in cosmology is noted.

•Aug 19, 2009•1 citation

Quantum Mechanics and Special Relativity: The Origin of Momentum Operator

Gintautas Kamuntavičius

The free relativistic particle, by definition, has to move in an inertial reference frame with uniform velocity less than the speed of light. The corresponding movement of a material quantum particle describes a wave packet, composed of matter waves—solutions of the Schr?dinger equation. The maximum of packet, corresponding to the largest probability to find the particle, has to move with the same uniform velocity, defined by the initial condition. It has been shown that the traditional definition of the quantum momentum operator i.e. taking it to correspond to the special relativity theory, relativistic momentum, cannot produce precise description of a relativistic matter particle. Different definitions are investigated and one that solves this issue is found. Obtained original expression of relativistic kinetic energy operator creates new possibilities for relativistic quantum systems theory.

Journal of Applied Mathematics and Physics•Jan 1, 2018•0 citations

Remarks on Relativistic Quantum Mechanics

K. V. Roberts

Physical Review•Jan 1, 1951•2 citations
A self-adjoint dynamical time operator is introduced in Dirac's relativistic formulation of quantum mechanics and shown to satisfy a commutation relation with the Hamiltonian analogous to that of the position and momentum operators. The ensuing time-energy uncertainty relation involves the uncertainty in the instant of time when the wave packet passes a particular spatial position and the energy uncertainty associated with the wave packet at the same time, as envisaged originally by Bohr. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie's hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli's objection is shown to be resolved or circumvented. Possible relevance to current developments in electron channeling, in interference in time, in Zitterbewegung-like effects in spintronics, graphene and superconducting systems and in cosmology is noted.
A self-adjoint dynamical time operator is introduced in Dirac’s relativistic formulation of quantum mechanics and shown to satisfy with the Hamiltonian a commutation relation analogous to that of the position and momentum operators. The instantaneous rate of change of the position expectation value with respect to the simultaneous expectation value of the dynamical time operator is shown to be the phase velocity, in agreement with de Broglie’s hypothesis of a particle associated wave whose phase velocity is larger than c. Thus, these two elements of the original basis and interpretation of quantum mechanics are integrated into its formal mathematical structure. Pauli’s objection is shown to be resolved or circumvented. Possible relevance to current developments in interference in time and in cosmology is noted.
The free relativistic particle, by definition, has to move in an inertial reference frame with uniform velocity less than the speed of light. The corresponding movement of a material quantum particle describes a wave packet, composed of matter waves—solutions of the Schr?dinger equation. The maximum of packet, corresponding to the largest probability to find the particle, has to move with the same uniform velocity, defined by the initial condition. It has been shown that the traditional definition of the quantum momentum operator i.e. taking it to correspond to the special relativity theory, relativistic momentum, cannot produce precise description of a relativistic matter particle. Different definitions are investigated and one that solves this issue is found. Obtained original expression of relativistic kinetic energy operator creates new possibilities for relativistic quantum systems theory.