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

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.