A quantum route to Hamilton–Jacobi equation: comments and remarks
We consider the short-wave limit of evolutionary wave equations to derive the Hamilton–Jacobi equation. In particular we consider how to get wave mechanics from the abstract picture on Hilbert spaces. This ‘realisation’ contains an additional ingredient which resembles the classical configuration space. We comment on the fact that Hamilton–Jacobi theory only considers one of the two equations obtained in the appropriate limit. We also comment on the fact that the superposition rule for the linear equation is no longer available for the approximate equation.