Determined position dependent Mass of the Rosen-Morse potential and its Bound state

Abstract

Abstract. In this paper, Schrödinger equation has been solved analytically with position dependent mass by the Rosen-Morse potential. The position dependent mass defined as the function 1/(1 − tanh(ηx)). Then corresponding position dependent mass substituted into Schrödinger equation. After that the obtained equation compared with associated Jacobi differential equation. Therefore, the eigenvalue and eigenfunction have been calculated, and from there the bound state has been found in terms of quantum numbers and the coefficients of potential. Also, the wave function has been obtained in terms of the associated Jacobi equation by µ = −3 and ν = 2. IntroductionOne of the fundamental wave equations is the Schrödinger equation that one used in physics and chemistry. Solutions of Schrödinger equation for some physical potential have important applications in molecular physics, quantum chemistry, nuclear, condensed matter physics, high energy physics and particle physics. These potentials are such as, and Woods-Saxon [8]. We usually solve these potentials as analytically and we can calculate energy eigenvalues and the corresponding wave functions exactly. Application of the mentioned potentials and Woods-Saxon potential are very useful to describe molecular structures and the interaction of a neutron with a heavy nucleus, respectively.Exact solutions of Schrödinger equation for the mentioned potentials are interesting in the fields of material science and condense matter physics. We note that, there are various methods to obtain exact solutions energy eigenvalues and corresponding wave function. One of the methods is a factorization method that can factorize the Hamiltonian of corresponding system in terms of production of two first order differential operators, in which so-called lowering and raising operators. The two obtained Hamiltonian are partner of each other. This method allowed to classify the problems according to the characteristics involved in the potentials and it is closely related to the supersymmetric quantum mechanics [9,10,11].In quantum mechanics, all of the analytically solvable potentials have a feature that be called as shape invariance. In fact, the shape invariance is an integrability condition and an interesting feature of supersymmetric quantum mechanics. This method is an exact and elegant technique to determine the eigenvalues and eigenfunctions of quantum mechanical problems [9].

Path integral solution for a deformed radial Rosen–Morse potential

“…Since then, it has attracted a lot of interest due to its numerous applications in several branches of physics [2,3]. It has also been used as an illustrative example in different methods such as the factorization method [4,5], the prepotential approach [6], the path integral technique [7,8,9,10], the supersymmetry in quantum mechanics and the shape invariance [11,12] and the Nikiforov-Uvarov method [13,14,15].…”