The Hamilton-Jacobi Theory

Abstract

The principal object of classical dynamics is to find where everything is at time t; that is, to find a set of q _n (t). The principal object of quantum mechanics is to find a wave function ψ(q, t). From this you cannot calculate where everything is in the sense of classical physics, but you can calculate all there is to know. Lagrangian and Hamiltonian dynamics find q(t) and q(t) or p(t) by means of ordinary differential equations. In ψ(q, t), q is in no sense a function of t, since ψ has a value for every q and every t. The equation it solves is a partial differential equation. The differences between the two mathematical descriptions are so wide that they seem to belong to different universes of ideas, but in Ehrenfest’s theorems (Sect. 1.3), for example, familiar ordinary differential equations came out of the partial differential equation for ψ(q, t) that are the same as the equations of Newtonian dynamics. It is now our task to show that out of the dynamics of Lagrange and Hamilton comes a partial differential equation that is the same as the eikonal equation (1.31), which was derived from wave optics. In Chap. 7 we will see how to reconstruct quantum mechanics, under suitable assumptions, if the eikonal equation is known. These arguments will help to explain the mathematical relation between the two theories, and a study of the necessary assumptions will do much to illuminate the subtler question of their physical relation.