Transition states and the critical parameters of central potentials

TL;DR

This paper investigates the properties of transition states at zero energy in quantum systems with central potentials, developing exact and approximation methods to determine critical parameters and energy level orderings.

Contribution

It introduces two exact methods and WKB approximations to calculate critical parameters for various central potentials, providing new analytic expressions and insights into bound state properties.

Findings

Derived asymptotic expressions for critical parameters as quantum numbers approach infinity.

Determined the maximum orbital quantum number for bound states in given potentials.

Revealed that energy level ordering depends on potential singularity at the origin.

Abstract

Transition states or quantum states of zero energy appear at the boundary between the discrete part of the spectrum of negative energies and the continuum part of positive energy states. As such, transition states can be regarded as a limiting case of a bound state with vanishing binding energy, emerging for a particular set of critical potential parameters. In this work we study the properties of these critical parameters for short range central potentials. To this end we develop two exact methods and also utilize the first and second order WKB approximations. Using these methods we have calculated the critical parameters for several widely used central potentials. The general analytic expressions for the asymptotic representations of the critical parameters were derived for cases where either the orbital quantum number $l$ or the number $n$ of bound states approaches infinity. The above mathematical models enable us to answer the following physical (quantum mechanical) questions: i) what is the number of bound states for a given central potential and given orbital quantum number $l$; ii) what is the maximum value of $l$ which can provide a bound state for the given central potential; iii) what is the order of energy levels for the given form of the central potential. It is revealed that the ordering of energy levels depends on the potential singularity at the origin.