Derivation of the Time-Dependent Schr\"odinger Equation from the Time-Independent One

TL;DR

This paper derives the time-dependent Schrödinger equation from the time-independent form by introducing a clock coordinate, highlighting its validity mainly for semiclassical clocks and discussing implications for physical interpretation.

Contribution

It presents a novel derivation of the time-dependent Schrödinger equation from the time-independent form using a clock coordinate, clarifying conditions for its applicability.

Findings

Standard time-dependent Schrödinger equation derived for semiclassical clocks

Different clock types analyzed for physical meaning

Formulas for density matrix and operator mean values obtained

Abstract

A derivation of the time-dependent Schr\"odinger equation from the time-independent one is considered. Instead of time, the coordinate of an additional degree of freedom, the clock, is introduced into the original time-independent Schr\"odinger equation. It is shown that the standard time-dependent Schr\"odinger equation can be obtained for the semiclassical clock only. For elucidation of the physical meaning of the equation obtained in this way, various types of clocks are discussed. In addition, the corresponding equation for the density matrix and formulas for the mean values of operators are derived.