Position‐dependent mass Schrödinger equations allowing harmonic oscillator (HO) eigenvalues

Quantum chemical systems with a position‐dependent mass have attracted the attention due to their relevance in describing the features of many microstructures of current interest. In this work, the point canonical transformation method applied to Schrödinger equations with a position‐dependent mass (SEPDM) is presented. Essentially, the proposal is aimed to transform the Schrödinger equation with a position‐dependent mass into a standard Schrödinger‐like equation for constant mass in such a way that the position‐dependent mass distribution (PDMD) becomes incorporated into the effective potential. As an useful application of the proposal, it is considered as effective potential the one‐dimensional harmonic oscillator potential model, which leads to those isospectral potentials related to different forms of PDMD. For example, the exactly solvable isospectral potentials involved in the SEPDM for some PDMD such as $2m(x) = e^{ - \alpha ^2 x^2 }$ , $1/(\alpha ^2 x^2 + 1)$ , $\exp (2\alpha x)/\mathop {\cosh }\nolimits^2 (\alpha x)$ , $1/\mathop {\cos }\nolimits^2 (\alpha x)$ , $\exp ( - \alpha |x|)$ , x α , and ${1 \over {b^2 }}\left( {{{b + \alpha ^2 x^2 } \over {1 + \alpha ^2 x^2 }}} \right)^2$ , are worked out explicitly including their raising and lowering operators that factorize the SEPDM for each PDMD allowing HO eigenvalues. However, the proposal is general and can be straightforwardly applied to other effective potential models as well as other PDMD that could be useful in quantum chemical applications. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008