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American Journal of Physics

Correlated wavefunction of two particles in an infinite well with a delta repulsion

José Rachid Mohallem, Leonardo Oliveira

Published Jun 1, 1990• 3 citations
Publisher

Abstract

The generator coordinate method is applied to the one-dimensional model problem of two particles in an infinite potential well with a delta-function repulsion potential between them. The aim is to produce a simple example involving correlated wavefunctions, accessible to undergraduate and graduate students in the area of quantum mechanics.

DOI
10.1119/1.16409

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The generator coordinate method is applied to the one-dimensional model problem of two particles in an infinite potential well with a delta-function repulsion potential between them. The aim is to produce a simple example involving correlated wavefunctions, accessible to undergraduate and graduate students in the area of quantum mechanics.
Abstract Exploring many-body interactions in an inquiry-based environment can enhance undergraduate and graduate student understanding of quantum physics. Specific classroom examples are provided in increasing complexity of content and maturity of comprehension. The simplest example of a purely quantum system (without using a classical potential) involves the many-body treatment of two quantum objects reflecting from each other via hard-sphere scattering in one dimensional motion. This example is extended to a particle confined in an infinite square well or interacting with a finite barrier and finite well where each of the objects generating the potentials now recoils due to reflection of the particle. These simple systems introduce concepts of entanglement, quantum correlated interference, and measurements made on a subset of the bodies. Completely inelastic scattering that involves the fusing (or destruction) of two bodies into the creation of a third is also discussed. In addition to this content, guided inquiry-based pedagogy is illustrated using two examples: collective recoil when a particle reflects from two scatterers and measurements on each body made asynchronously. The examples provided explore potential methods to verify quantum effects on a macroscopic scale and address measurement in a quantum system.
The problem of one-dimensional particles interacting via a delta-function potential has been a useful model for the many-body problem. While it can be solved for bosons, it has so far not been solved for fermions except for the special cases of either all spins parallel or all but one spin parallel. The full difficulty of the problem first manifests itself when two spins are down (i.e., $S=\frac{1}{2}N\ensuremath{-}2$), and this we solve here. We are confident that our method can be extended to the general problem.
After briefly reviewing the definitions of classical probability densities for position, P CL ( x ), and for momentum, P CL ( p ), we present several examples of classical mechanical potential systems, mostly variations on such familiar cases as the infinite well and the uniformly accelerated particle for which the classical distributions can be easily derived and visualized. We focus especially on a simple potential which interpolates between the symmetric linear potential, V ( x ) = F | x |, and the infinite well, which can illustrate, in a mathematically straightforward way, how the divergent δ-function classical probability density for momentum for the infinite well can be seen to arise. Such examples can help students understand the quantum mechanical momentum-space wavefunction (and its corresponding probability density) in much the same way that other semiclassical techniques, such as the WKB approximation, can be used to visualize position-space wavefunctions.