A New Solvable Generalized Trigonometric Tangent Potential Based on SUSYQM

Supersymmetric quantum mechanics has wide applications in physics. However, there are few potentials that can be solved exactly by supersymmetric quantum mechanics methods, so it is undoubtedly of great significance to find more potentials that can be solved exactly. This paper studies the supersymmetric quantum mechanics problems of the Schrodinger equation with a new kind of generalized trigonometric tangent superpotential: Atan npx + B tan mpx. We will elaborate on this new potential in the following aspects. Firstly, the shape invariant relation of partner potential is generated by the generalized trigonometric tangent superpotential. We find three shape invariance forms that satisfy the additive condition. Secondly, the eigenvalues and the eigenwave functions of the potential are studied separately in these three cases. Thirdly, the potential algebra of such a superpotential is discussed, and the discussions are explored from two aspects: one parameter's and two parameters' potential algebra. Through the potential algebra, the eigenvalue spectrums are given separately which are consistent with those mentioned earlier. Finally, we summarize the paper and give an outlook on the two-parameter shape-invariant potential.

Automated summary

This paper studies the supersymmetric quantum mechanics problems of the Schrodinger equation with a new kind of generalized trigonometric tangent superpotential. It provides a shape invariant relation for partner potential and explores the eigenvalues and eigenwave functions in different cases. It also discusses the potential algebra of such a superpotential and gives an outlook on the two-parameter shape-invariant potential.