Wave Mechanics

Abstract

The Hamilton-Jacobi description of a system of particles with mass m _i and kinetic energy T _i = P ² _i /2m _i permits the interpretation of the real trajectories of such particles as lines orthogonal to the surfaces S (q, $\vec \nabla$ W, t. By taking these surfaces to analogous to optical wave surfaces, it is possible to introduce an “equation of the mechanical wave” or wave mechanics describing the evolution of a wave function ψ ($\vec r$, t) with frequency given via E = h v and with wavelength λ = h/p. This wave function, which plays the role of the optical wave $\vec E$ or $\vec H$, obeys a fundamental equation of propagation, the Schrödinger equation, written from the Hamiltonian formulation of a classical system using a correspondence principle between the classical variables momentum, $\vec E$, and energy, E, and operators acting on the wave function sought.