Abstract
In this paper, a new method for the exact solution of the stationary, one-dimensional Schrödinger equation is proposed. Application of the method leads to a three-parametric family of exact solutions, previously known only in the limiting cases. The method is based on solutions of the Ricatti equation in the form of a quadratic function with three parameters. The logarithmic derivative of the wave function transforms the Schrödinger equation to the Ricatti equation with arbitrary potential. The Ricatti equation is solved by exploiting the particular symmetry, where a family of discrete transformations preserves the original form of the equation. The method is applied to a one-dimensional Schrödinger equation with a bound states spectrum. By extending the results of the Ricatti equation to the Schrödinger equation the three-parametric solutions for wave functions and energy spectrum are obtained. This three-parametric family of exact solutions is defined on compact support, as well as on the whole real axis in the limiting case, and corresponds to a uniquely defined form of potential. Celebrated exactly solvable cases of special potentials like harmonic oscillator potential, Coulomb potential, infinite square well potential with corresponding energy spectrum and wave functions follow from the general form by appropriate selection of parameters values. The first two of these potentials with corresponding solutions, which are defined on the whole axis and half axis respectively, are achieved by taking the limit of general three-parametric solutions, where one of the parameters approaches a certain, well-defined value.
Highlights
One of the main open problems in nonrelativistic quantum mechanics is finding exact bound states to microscopic systems
Celebrated exactly solvable cases of special potentials like harmonic oscillator potential, Coulomb potential, infinite square well potential with corresponding energy spectrum and wave functions follow from the general form by appropriate selection of parameters values
This paper introduces a three-parametric family of potentials that encompasses, as special cases, all of the potentials mentioned in the previous paragraph
Summary
One of the main open problems in nonrelativistic quantum mechanics is finding exact bound states to microscopic systems. The methods to solve the Schrödinger equation using continued fractions are utilized since the 1970s [22,23] In this contribution, continued fractions are used to solve the general form of the potential and allow one to find the exact solutions of the Schrödinger equation simultaneously, which previously was obtained only in the special, isolated cases [24]. The present work goes beyond the results of [25,26] This technique proves interesting from the point of view of new exactly solvable potentials and can be extended to the potentials that do not satisfy the shape invariance condition [27]
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