Time in quantum mechanics: A fresh look at the continuity equation

Abstract

The local conservation of a physical quantity whose distribution changes with time is mathematically described by the continuity equation. The corresponding time parameter, however, is defined with respect to an idealized classical clock. We consider what happens when this classical time is replaced by a non-relativistic quantum-mechanical description of the clock. From the clock-dependent Schrödinger equation (as analogue of the time-dependent Schrödinger equation) we derive a continuity equation, where, instead of a time-derivative, an operator occurs that depends on the flux (probability current) density of the clock. This clock-dependent continuity equation can be used to analyze the dynamics of a quantum system and to study degrees of freedom that may be used as internal clocks for an approximate description of the dynamics of the remaining degrees of freedom. As an illustration, we study a simple model for coupled electron-nuclear dynamics and interpret the nuclei as quantum clock for the electronic motion. We find that whenever the Born-Oppenheimer approximation is valid, the continuity equation shows that the nuclei are the only relevant clock for the electrons.

References in corpus (21)

• Exact factorization of the time-dependent electron-nuclear wavefunction
• Correlated electron-nuclear dynamics: Exact factorization of the molecular wavefunction
• Quantum Time
• Time from quantum entanglement: an experimental illustration
• Autonomous quantum clocks: does thermodynamics limit our ability to measure time?
• Evolution without evolution, and without ambiguities
• Quantizing Constrained Systems: New Perspectives
• Exact factorization-based density functional theory of electrons and nuclei
• The adiabatic limit of the exact factorization of the electron-nuclear wave function
• Time Dependence in Quantum Mechanics
• System-time entanglement in a discrete time model
• The equivalent emergence of time dependence in classical and quantum mechanics
• Quantum Time: experimental multi-time correlations
• Quantum heat transfer: A Born Oppenheimer method
• Internal clock formulation of quantum mechanics
• On Factorization of Molecular Wavefunctions
• Realistic clocks for a Universe without time
• Investigating the origin of time with trapped ions
• A minimalist approach to conceptualization of time in quantum theory
• The time as an emergent property of quantum mechanics, a synthetic description of a first experimental approach
• Born-Oppenheimer description of two atoms in a combined oscillator and lattice trap

Cited by in corpus (7)

• Timelessness strictly inside the Quantum Realm
• Many-electron effects of strong-field ionization described in an exact one-electron theory
• Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology
• Emergent dynamics from entangled mixed states
• Electronic quantum trajectories with quantum nuclei
• A link between static and dynamical perturbation theory
• Time in quantum mechanics: A fresh look on quantum hydrodynamics and quantum trajectories