Abstract
In this paper, first we introduce the three dimensional non-linear oscillator with a position dependent mass. In that case we start by the stationary Schrodinger equation which is generated by the three dimensional Hamiltonian. The wave function depend on three spatial variables and the usual process of variable in spherical coordinates and the wave functions will be of radial and angular solutions. We can easily solve the angular part of equation but redial part of equation will be complicated. In that case, we take advantage from sl(2) algebra and write the corresponding equation in terms of P+(r), P-(r) and P0(r) which are generators of generalized sl(2) algebra. The information of this algebra help us to obtain the energy spectrum and wave function from radial part of equation.
Highlights
The one dimensional non-linear quantum oscillator has been studied by Ref.[1]
In one dimension for the quantum mechanical version of model admits exact solution for the wave functions and the energy eigenvalues [1]. This solution was achieved by using the factorization method and solution of wave function represented by Rodrigues - type of formula
One dimensional non-linear quantum oscillator model was farther studied by Ref.[2], and have shown that the exact solution are expressed in closed - form in terms of special function, without to use the factorization method or Rodrigues formula
Summary
The one dimensional non-linear quantum oscillator has been studied by Ref.[1] This model show us two important things in physics, the first one is non-linearity of the potential V (x) and second one is position dependent mass M (x). In one dimension for the quantum mechanical version of model admits exact solution for the wave functions and the energy eigenvalues [1]. One dimensional non-linear quantum oscillator model was farther studied by Ref.[2], and have shown that the exact solution are expressed in closed - form in terms of special function, without to use the factorization method or Rodrigues formula. The commutation relation of such operators with comparing to usual sl(2) algebra help us to achieve the energy spectrum and radial wave function
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