The quantum effective mass Hamilton–Jacobi problem

Abstract

Abstract. In this article, the quantum Hamilton-Jacobi theory based on the position dependent mass model is studied. Two effective mass functions having different singularity structures are used to examine the Morse and Pöschl-Teller potentials. The residue method is used to obtain the solutions of the quantum effective mass-Hamilton Jacobi equation. Further, it is shown that the eigenstates of the generalized non-Hermitian Swanson Hamiltonian for Morse and Pöschl-Teller potentials can be obtained by using the Riccati equation without solving a differential equation.

Quantum Hamilton–Jacobi quantization and shape invariance

“…Depending on values of λ and signs of a, it produces different conventional potentials. We will now consider the case 17 with a < 0 and λ = 1. The needed boundary conditions for W at the left and the right boundaries also require that we have B < 0.…”

“…QHJ quantization is an elegant method to determine eigenspectra for quantum mechanical systems from the singularity structure of the underlying potential. On case-by-case basis, many researchers have demonstrated the efficacy of this formalism to determine eigenvalues and eigenfunctions for a multitude of potentials [12][13][14][15][16][17][18][19][20][21][22][23]. In [24], the authors showed that QHJ and the SI condition do help determine the spectrum for superpotentials that are either algebraic or exponential functions of the coordinate in unbroken supersymmetric phase; the general form of all conventional potentials were not known at that point.…”

Dual wavefunctions in two–dimensional quantum mechanics

“…The theory described by Eqs. ( 3) and ( 4), is also called Quantum Hamilton-Jacobi theory, and has been extensively studied in the context of their differences with respect to Hamilton-Jacobi equations [2][3][4][5][6][7][8][9][10].…”

Exact treatment of the relativistic double ring-shaped Kratzer potential using the quantum Hamilton–Jacobi formalism

“…6,7 Also, this method was used to study quasi-exactly solvable models 8 and to obtain analytical solutions for two-dimensional central potentials, 9 for two-dimensional singular oscillator, 10 supersymmetric potentials, 11 non-Hermitian exponential-type potentials, 12 PT symmetric Hamiltonians 13,14 and the position-dependent mass problem. 15 The working methodology of this formalism is mainly based on the knowledge of the singularity structure of the quantum momentum function (QMF) defined as the logarithmic derivative of the wave function. 1,3 In one dimension, the QMF is a solution of the following differential equation:…”