On Exactly Solvable Potentials

We investigate two methods of obtaining exactly solvable potentials with analytic forms. PACS numbers:03.65.Ge,11.30.Pb There are two methods of obtaining exactly solvable potentials in quantum mechanics. The first method was developed by applying the technique of supersymmetry (SUSY) to the Schrödinger equation and obtain two potentials with almost identical spectra. The two potentials can be considered to be superpartners of each other. It has been shown [1, 2] that if the two partners happen to be related by a simple relationship called shape invariance, the energy eigenvalues of the potential can be solved exactly. All the known solvable potentials with closed analytic forms can be shown to be shape invariant. In the literature, there are other solvable potentials that have not been shown to be shape invariant, however, they exist only complex numerical forms which we shall not consider in this article. A second interesting method of obtaining solvable potential was proposed by Klein and Li[3] based on some special quantum commutation relationships. Li[4] has recently worked out the most general potential that can be obtained this way. In this paper, we first investigate the relationship between the two approaches. We will show that the general solutions obtained by Li are special cases of the general solutions that can be obtained by solving shape invariance condition. Therefore, the solving shape invariance condition remains the most general method of obtaining exactly solvable potentials which can be given in analytic form. Unfortunately, there is no general method for getting analytic solutions of shape invariance condition. The best one can do seems to be starting from a guessing ansatz with an unknown function. The shape invariance is then enforced by demanding that the function satisfies an ordinary differential equation. The ansatz allows one to turn the difficult shape invariance condition into a problem of solving differential equation. It also allows one to associate each ansatz as defining a particular class of solutions. While the solutions obtained by Li correspond to those defined by a particular ansatz, in the literature, the largest classes of analytic solutions of the shape invariance condition was provided by Gendenshtein [1]. He proposed three ansatzs which define three classes of solutions which seem to cover all the known analytic exactly solvable potentials. We, therefore, proceed to solve the differential equations corresponding to these ansatzs and the energy eigenvalues of these three classes of solutions of shape invariance condition. In the process, we also demonstrate that all the known