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 timedependent Schrödinger equation) we derive a continuity equation, where, instead of a timederivative, 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 aquantum 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.

Gauge Fixing and the Semiclassical Interpretation of Quantum Cosmology

“…One may verify that this equation is invariant under quantum background transformations by using (31), (70) and (73). This is also understood from the fact that (74) is simplyĤ ϕ Ψ ϕ = 0.…”

“…as ∂ ∂t := −i M χ n a=1 ∂χ ∂Qa ∂ ∂Qa . Such a notion of "quantum time" was considered in [32,70]. However, since the choice of χ is arbitrary due to equation (4), the definition of t from the phase ϕ is sufficient.…”

Time in quantum theory, the Wheeler–DeWitt equation and the Born–Oppenheimer approximation

“…This topic now attracts much attention by researchers working in areas which are at first glance rather distant from quantum gravity and cosmology. In a recent preprint [15] a rather detailed review devoted to the appearance of time in quantum mechanics is presented. The main idea is the following: let us consider a quantum system consisting of two subsystems, whose wave function satisfies a time-independent Schrödinger equation with a fixed value of energy.…”

“…The expression |χ(R)| 2 gives the marginal probability density for the quantum clock and |φ(r|R)| 2 gives the conditional probability for the system under consideration [16,17]. Then, performing the operations which are usually connected with the Born-Oppenheimer approach, one can obtain the so called clock-dependent Schrödinger equation [15] for the subsystem under consideration, having the form…”

“…(41) behaves as a partial derivative with respect to the classical time. In the approach reviewed in the paper [15] one can underline two important features: first, the exact factorisation of the wave function (40) is always possible and it is always possible to obtain the clock-dependent Schrödinger equation from the time-independent Schrödinger equation. Second, a quantum clock does not always give rise to (semi)-classical time.…”