# Symmetric Fermi-type potential

## TL;DR

This paper analyzes a symmetric Fermi-type potential in quantum mechanics, deriving conditions for bound states and providing a semi-classical formula to predict the number of bound states, validated against neutron energy levels.

## Contribution

It introduces a semi-classical expression for counting bound states in a symmetric Fermi potential and confirms its accuracy with neutron energy level data.

## Key findings

- Derived conditions for n-node half-bound states at zero energy.
- Established a semi-classical formula for the number of bound states.
- Validated the formula with neutron energy level data.

## Abstract

We utilize the amenability of the Fermi-type potential profile in Schr{\"o}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 $\alpha=a/b, ~\beta_n {=}b\sqrt{2m U_n}/\hbar$, we find $\beta_n$ 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 ${\cal G}(\alpha,\beta)$ such that the Fermi well has either $[\cal G]$ or $[{\cal G}]+1$ number of bound states. Here $[.]$ indicates the integer part. We also confirm the consistency of $\cal G$ with the number of s-wave neutron energy levels in a central ($x\in (0,\infty))$ Fermi potential well.