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Chinese Journal of Physics

Nontrivial Exactly Solvable Potentials with Linear Equations of Motion

Ching‐Teh Li

Published Aug 1, 1994• 1 citation
OpenAlex

Abstract

Based on an equations-of-motion method in the framework of Heisenberg's matrix mechanics, we investigate the conditions under which a one-dimensional quantum mechanical system becomes exactly solvable. By linearizing the equation of motion which is a double-commutation relation of some appropriately chosen function of the position operator with the Hamiltonian, we obtained a set of nontrivial exactly solvable potentials in one dimension. These potentials not only can be solved analytically in closed forms but also contain both the Morse potential and the Poschl-Teller potential as their limiting cases. They may thus be valuable for some potential model calculations as well as for testing various approximation schemes. We also examine these potentials in the framework of supersymmetric quantum mechanics, which is particularly useful for studying exactly solvable potentials.

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Based on an equations-of-motion method in the framework of Heisenberg's matrix mechanics, we investigate the conditions under which a one-dimensional quantum mechanical system becomes exactly solvable. By linearizing the equation of motion which is a double-commutation relation of some appropriately chosen function of the position operator with the Hamiltonian, we obtained a set of nontrivial exactly solvable potentials in one dimension. These potentials not only can be solved analytically in closed forms but also contain both the Morse potential and the Poschl-Teller potential as their limiting cases. They may thus be valuable for some potential model calculations as well as for testing various approximation schemes. We also examine these potentials in the framework of supersymmetric quantum mechanics, which is particularly useful for studying exactly solvable potentials.
The equation-of-motion method based on Heisenberg's matrix mechanics, utilized in this paper, was originally developed within the context of the nuclear many-body problem. The method was later applied successfully to anharmonic oscillators and other problems in quantum mechanics with polynomial potentials in one and two dimensions. Here we apply the method to the one-dimensional quantum system with the nonpolynomial potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ as an illustration of its applicability to a wider class of problems than has been previously considered. We examine some of the structural and numerical aspects of this one-dimensional quantum system, and show that it is essential to work with sinhx\mathrm{sinh}xsinhx instead of x as the dynamical variable for this nonpolynomial potential problem. This choice is suggested by a preliminary study of the exactly solvable Poschl-Teller potential, $\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,$ where our methods provide an exact algebraic solution. For the potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.
We extend the notion of Dirac oscillator in two dimensions, to construct a set of potentials. These potentials become exactly and quasi-exactly solvable potentials of nonrelativistic quantum mechanics when they are transformed into a Schrödinger-like equation. For the exactly solvable potentials, eigenvalues are calculated and eigenfunctions are given by confluent hypergeometric functions. It is shown that, our formulation also leads to the study of those potentials in the framework of the supersymmetric quantum mechanics.
Searching for non-Hermitian, -symmetric Hamiltonians [1] with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a symmetric non-Hermitian Hamiltonian model which is given as where ω and α are real constants, and are first order differential operators. Because the Hamiltonian is pseudo-Hermitian, we have obtained the Hermitian equivalent of which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened and hyperbolic Rosen-Morse II potentials. Using convenient sinilarity transformations, we have obtained a physical Hamiltonian h for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics [2].
A general algebraic procedure for constructing coherent states of a wide class of exactly solvable potentials, e.g., Morse and P\"oschl-Teller, is given. The method, a priori, is potential independent and connects with earlier developed ones, including the oscillator-based approaches for coherent states and their generalizations. This approach can be straightforwardly extended to construct more general coherent states for the quantum-mechanical potential problems, such as the nonlinear coherent states for the oscillators. The time evolution properties of some of these coherent states show revival and fractional revival, as manifested in the autocorrelation functions, as well as, in the quantum carpet structures.
Supersymmetrical intertwining relations of second order in derivatives allow to construct a two-dimensional quantum model with complex potential, for which {\it all} energy levels and bound state wave functions are obtained analytically. This model {\it is not amenable} to separation of variables, and it can be considered as a specific complexified version of generalized two-dimensional Morse model with additional sinh⁡−2\sinh^{-2}sinh−2 term. The energy spectrum of the model is proved to be purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. The symmetry operator is found, the biorthogonal basis is described, and the pseudo-Hermiticity of the model is demonstrated. The obtained wave functions are found to be common eigenfunctions both of the Hamiltonian and of the symmetry operator.