Symmetric Fermi-type potential

Abstract

We utilize the amenability of the Fermi-type potential profile in Schr{ö}dinger equation to construct a symmetric one dimensional well as $V(x)=-U_n/[1+\exp [(|x|-a)/b]],U_n=V_n[1+\exp [-a/b]]$. We define $\hat{I}\pm =a/b,{\hat{I}}_n^2=b\sqrt{2m{U}_n}/\hslash$, we find ${\hat{I}}_n^2$ values for which critically the well has $n$-node half bound state at $E=0$. Consequently, this fixed well has $n$ number of bound states. Also we obtain a semi-classical expression $\mathcal{G}(\hat{I}\pm ,{\hat{I}}^2)$ such that the Fermi well has either $[\mathcal{G}]$ or $[\mathcal{G}]+1$ number of bound states. Here $[.]$ indicates the integer part. We also confirm the consistency of $\mathcal{G}$ with the number of s-wave neutron energy levels in a central ($x\in (0,\infty ))$ Fermi potential well.

Seven pages, 3 Figures and 3 Tables some new Refs. added, To appear in Am. J. Phys