Breakdown of separability due to confinement, submitted to Report on mathematical physics

TL;DR

This paper investigates how confinement and boundary conditions in a two-particle system in two dimensions lead to the breakdown of separability in the Schrödinger equation, reducing the problem to a single particle in a limited domain.

Contribution

It demonstrates the reduction of a two-particle problem with boundary constraints to an effective single-particle problem in a confined domain, using Green's functions and specific geometries.

Findings

Separable Schrödinger equations can be broken down under confinement conditions.

Green's functions with Jacobi theta functions effectively model confined states.

Confinement alters the reducibility of multi-particle quantum systems.

Abstract

A simple system of two particles in a bidimensional configurational space $S$ is studied. The possibility of breaking in $S$ the time independent Schr\"{o}dinger equation of the system into two separated one-dimensional one-body Schr\"{o}dinger equations is assumed. In this paper, we focus on how the latter property is countered by imposing such boundary conditions as confinement in a limited region of $S$ and/or restrictions on the joint coordinate probability density stemming from the sign-invariance condition of the relative coordinate (an impenetrability condition). Our investigation demonstrates the reducibility of the problem under scrutiny into that of a single particle living in a limited domain of its bidimensional configurational space. These general ideas are illustrated introducing the coordinates $X_c$ and $x$ of the center of mass of two particles and of the associated relative motion, respectively. The effects of the confinement and the impenetrability are then analyzed by studying with the help of an appropriate Green's function and the time evolution of the covariance of $X_c$ and $x$. Moreover, to calculate the state of the single particle constrained within a square, a rhombus, a triangle and a rectangle the Green's function expression in terms of Jacobi $\theta_3$-function is applied. All the results are illustrated by examples.