The solution of the one-dimensional Schrödinger equation is discussed in the case of position-dependent mass. The general formalism is specified for potentials that are solvable in terms of generalized Laguerre polynomials and mass functions that are positive and bounded on the whole real x axis. The resulting four-parameter potential is shown to belong to the class of "implicit" potentials. Closed expressions are obtained for the bound-state energies and the corresponding wave functions, including their normalization constants. The constant mass case is obtained by a specific choice of the parameters. It is shown that this potential contains both the harmonic oscillator and the Morse potentials as two distinct limiting cases and that the original potential carries several characteristics of these two potentials. Possible generalizations of the method are outlined.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics