It is shown that for nonrelativistic central potentials of specific forms, the bound state wavefunctions u (r) have the property that G (r) ≡ℱ r0dx u (x)(∂u (x)/∂μ) ⩽0 for all r. Here μ is the reduced mass. Thus, for such potentials, the probability that a particle lies within a spherical shell of radius r is a monotonically increasing function of μ. The forms for which this property has been established are (a) V (r) =Crε, −2<ε<∞; (b) V (ir) =C lnr, and (c) S waves in the potential, V (r) =−V0 (0⩽r<a), V (r) =0 (a⩽r<∞). For all monotonically increasing potentials, G (r) ≳0 for nodeless states. It is shown explicitly that G (r) cannot be nonnegative for all bound states in an arbitrary monotonically increasing potential. The question remains open regarding the widest class of potentials for which G (r) is nonnegative for all bound states.
The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.
The position-dependent mass Schrodinger equation (PDMSE) comes from the O von Roos quantum Hamiltonian which models to position-dependent effective mass quantum systems. The algebraic study of its factorization is compared in this work with the factorization of the traditional constant mass Schrödinger equation so that both equations are related by similarity transformations. The approach allows building solvable cases of the PDMSE for any value of the ambiguity parameters in the general Hamiltonian of O von Roos, so it can be considered as a unified treatment of the PDMSE that contains as particular cases those Hamiltonians of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li- Kuhn, among others. We explicitly show the PDMSE solutions coming from the harmonic, Morse and multiparameter exponential-type potentials. The method is general and can be easily extended to other potential models and position dependent mass distributions useful in the mathematical modeling of quantum systems.