The time dependent Schrodinger equation revisited I: quantum field and classical Hamilton-Jacobi routes to Schrodinger's wave equation

Abstract

The time dependent Schrodinger equation is frequently ‘derived’ by postulating the energy E → iℏ(∂/∂t) and momentum p⃗ → (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.