The Schrödinger Wave Equation

JOHN C. MORRISON

S. Rajasekar, R. Velusamy

During 1925 three different but equivalent versions of quantum theory were proposed, Schrodinger proposed wave mechanics; Heisenberg developed matrix mechanics and Dirac-introduced operator theory. Considering the de Broglie's matter waves Erwin Schrödinger, an Austrian physicist, argued that if a particle like an electron behaves as a wave then the equation of wave motion could be successfully applied to it. He postulated a function varying in both space and time in a wave-like manner. Schrodinger set up a linear and time-dependent wave-like equation, called Schrodinger wave equation, to describe the wave aspect of a particle taking account of de Broglie's relation for wavelength. This chapter discusses the Schrodinger equation and explores the general properties of the solution of the Schrodinger equation. It verifies the Ehrenfest's theorem in certain interesting systems and solves equation for linear harmonic oscillator and shows that their solutions are identical to the solutions of classical equation of motion.

John C. Morrison

Stanley M. Flatté

The use of the Schrödinger equation in classical physics was begun in 1950; in the classical context it is known as the parabolic wave equation. The derivation of the parabolic wave equation from the full classical wave equation is analogous to deriving the Schrödinger equation from the Klein–Gordon equation. An introduction to the Schrödinger equation by this route could be of help in introductory quantum-mechanics classes.

Marlan O. Scully

The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i ℏ(∂/∂ t ) and momentum → (ℏ/ i ) operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation.

John Morrison