Time-Dependent Schrödinger Equation

Abstract

The time-dependent Schrödinger equation is derived from the superposition principle, in the conservative case first, then in the general case. The derivation of the continuity equation follows, leading to the concept of wave packet and density of probability flux. Then, the wave packet for a free particle is investigated in detail, and the concept of group velocity is introduced. The first complement deals with an application of the semiclassical approximation; through it one explains why an electron belonging to a stationary state emits no power, namely, why the radiative decay predicted by the classical model does not occur. The polar form of the time-dependent Schrödinger equation is then shown, that brings about an interesting similarity with the Hamilton-Jacobi equation of Classical Mechanics. The last complement deals with the Hamiltonian operator of a particle subjected to an electromagnetic field, and shows the effect of a gauge transformation on the wave function.