Quantum Hamilton-Jacobi formalism and the bound state spectra

Abstract

It is well known in classical mechanics that, the frequencies of a periodic system can be obtained rather easily through the action variable, without completely solving the equation of motion. The equivalent quantum action variable appearing in the quantum Hamilton-Jacobi formalism, can, analogously provide the energy eigenvalues of a bound state problem, without having to solve the corresponding Schrödinger equation explicitly. This elegant and useful method is elucidated here in the context of some known and not so well known solvable potentials. It is also shown, how this method provides an understanding, as to why approximate quantization schemes such as ordinary and supersymmetric WKB, can give exact answers for certain potentials.