Quantum anharmonic oscillator plus delta-function potential: a molecular view of pairing formation and breaking in the coordinate space

Abstract

We propose an alternative way to describe the pairing formation and breaking via a quantum anharmonic oscillator with a delta-function potential model. Unlike BCS theory, which describes the pairing formation in the momentum space, this model works in the coordinate space and is able to give a molecular view of pairing formation and breaking in the coordinate space. By exploring the dynamical interplay between the intrinsic factor (dissociation energy) and external factor (pairing strength) of this system additional information was gained, including the critical pairing strength and critical scattering length, which might relate to the BCS–BEC crossover phenomena and halo state formation. Although only the energetic aspect of pairing is described by this model, its simplicity and pedagogical steps might help undergraduate students to understand the pairing problem in a simple way.

Reply to the comment on ‘Quantum anharmonic oscillator plus delta-function potential: a molecular view of pairing formation and breaking in the coordinate space’

“…The main feature of our numerical results is quite reasonable (and as expected) as indicated by the formation of a distorted (unsymmetrical) probability wave function. The equation ( 16) in our paper [2] is the anharmonic oscillator's version of equation ( 7) (or equation ( 5) in Fernandez's comment paper) and features a similar behavior as QHOD, except for the distorted probability wavefunctions as a result of the Morse potential.…”

“…We would like to thank Professor Fernandez for reminding us to cite Atkinson and Crater [1] in our paper [2] regarding the first analytical solution of the quantum harmonic oscillator with the addition of a delta potential (QHOD). There are plenty of papers discussing the solution of QHOD, and we have sampled and cited some of them in our paper i.e.…”

Simulating the Dynamics of Stock Price via PSO-Assisted Quantum Anharmonic Oscillator Model: Case of Jakarta Composite Index

“…The quantum harmonic and anharmonic oscillator model is perhaps the most used model to describe the dynamics of the system in many cases, such as atomic [9], [10], nuclear [11], [12], or even econophysics [13], [14] problems. Our study is a modification of Gao's work in Ref [14], with the addition of the PSO algorithm for parameter optimization and the local market (Jakarta Composite Index) as the data source.…”

Comment on ‘Quantum anharmonic oscillator plus delta-function potential: a molecular view of pairing formation and breaking in the coordinate space’

“…In a paper published recently in this journal [1] the authors solved the Schrödinger equation with an Hermitian operator H 0 perturbed by a delta-function potential −gδ(x), where δ(x) is the Dirac delta function. In their analysis H 0 is either the harmonic oscillator or the Morse oscillator.…”

“…The authors attributed this expression to a dissertation thesis [1], but the fact is that it is well known since long ago [2].…”