Shape Invariance and the Exactness of Quantum Hamilton-Jacobi Formalism

Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two parallel methods to determine the spectra of a quantum mechanical systems without solving the Schrödinger equation. It was recently shown that the shape invariance, which is an integrability condition in SUSYQM formalism, can be utilized to develop an iterative algorithm to determine the quantum momentum functions. In this paper, we show that shape invariance also suffices to determine the eigenvalues in Quantum Hamilton-Jacobi Theory. Quantum Hamilton-Jacobi Theory and supersymmetric quantum mechanics (SUSYQM) are two very different methods that give eigenvalues for quantum mechanical systems without solving the Schrödinger differential eigenvalue equation. Supersymmetric quantum mechanics is a generalization of Dirac’s ladder operator method for the harmonic oscillator. This method consists of factorizing Schrödinger’s second order differential operator into two first order differential operators that play roles analogous to ladder operators. If the interaction of a quantum mechanical system is described by shape invariant potentials [1, 2], SUSYQM allows one to generate all eigenvalues and eigenfunctions through algebraic methods. Another formulation of quantum mechanics, the Quantum Hamilton-Jacobi (QHJ) formalism, was developed by Leacock and Padgett [3] and independently by Gozzi [4]. It was made popular by a series of papers by Kapoor et. al. [5]. In this formalism one works with the quantum momentum function (QMF) p(x), which is related to the wave function ψ through the relationship p(x) = −ψ′(x)/ψ(x), where prime denotes differentiation with respect to x. Our definition of QMF’s differs by a factor of i ≡ √ −1 from that of ref. [3, 5, 6], where they define p(x) = −i ′(x) ψ(x) ; we use p(x) = − ψ′(x) ψ(x) . It was shown, on a case by case basis, that the singularity structure of the function p(x) determines the eigenvalues of the Hamiltonian [3, 4, 5] for all known solvable potentials. Kapoor and his collaborators have shown that the QHJ formalism can be used not only to determine the eigenvalues of the Hamiltonian of the system, but also its eigenfunctions [7]. They have also used QHJ to analyze Quasi-exactly solvable systems where only an incomplete set of the eigenspectra can be derived analytically and also to study periodic potentials [8]. It is important to note that all cases worked out in Refs. [3, 5] satisfied the integrability condition known as the translational shape invariance for which the SUSYQM method always gave the exact result. e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] Dirac attributes this method to Fock.