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

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 nonrelativistic quantum-mechanical description of the clock. From the clock-dependent Schrödinger equation (as an analog 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.

Figure 1

The two lowest Born-Oppenheimer potential energy surfaces of the proton-coupled electron transfer model as well as the initial nuclear densities ${\left|\chi \right|}^2$ for mass parameters ${\mu }^{-1}=100$ and 900 are shown for a case of weak coupling (left, $R_{\mathrm{c}}=4.0$ $a_0$) and strong coupling (right, $R_{\mathrm{c}}=7.0$ $a_0$).

Figure 2

Measures $N_t$, $N_c$, $N_J$ of the three contributions occurring in the continuity equation for the electron density for a mass parameter ${\mu }^{-1}=900$, for different coupling parameters $R_{\mathrm{c}}$.

Figure 3

Measures $N_t$, $N_c$, $N_J$ of the three contributions occurring in the continuity equation for the electron density for a coupling parameter $R_{\mathrm{c}}=4.0$ $a_0$, for different mass parameters ${\mu }^{-1}$.