General approach to potentials with two known levels
Abstract
Description
We present the general form of potentials with two given energy levels $E_{1}$, $E_{2}$ and find corresponding wave functions. These entities are expressed in terms of one function $ξ(x)$ and one parameter $ΔE=E_{2}$-$E_{1}$. We show how the quantum numbers of both levels depend on properties of the function $ξ(x)$. Our approach does not need resorting to the technique of supersymmetric (SUSY) quantum mechanics but automatically generates both the potential and superpotential.
14 pages, REVTeX 3.0. In v.2 misprints and inaccuracies in presentation corrected, discussion of 3-dim. case added. In v.3 misprint in eq. 41, several typos and inaccuracies in English corrected. To be published in J. of Phys. A: Math. Gen
14 pages, REVTeX 3.0. In v.2 misprints and inaccuracies in presentation corrected, discussion of 3-dim. case added. In v.3 misprint in eq. 41, several typos and inaccuracies in English corrected. To be published in J. of Phys. A: Math. Gen