State change via one-dimensional scattering in quantum mechanics

TL;DR

The work investigates state change via one-dimensional scattering in a two-particle system where a free particle interacts with a particle confined to a box through a local delta potential, leading to a finite set of outcome states constrained by energy conservation. By reformulating the problem as an infinite system of Fredholm integral equations for the scattered amplitudes and solving numerically with second-order accuracy, the authors compute the asymptotic probabilities for the bound particle to occupy discrete box states, along with the left/right scattering probabilities of the free particle. The method is nonperturbative and valid for arbitrary interaction strength, contrasting with the Born approximation which diverges or becomes inaccurate at stronger couplings; it also reveals rich structure in the outcome probabilities, including quasi-bound-state effects and strong dependence on dimensionless parameters. The approach offers a robust framework for analyzing discrete state changes in quasi-one-dimensional quantum systems, with potential relevance to quantum wires and quantum dots, and is accompanied by publicly available code.

Abstract

This study aims to address the nature of state change, measurement, and probabilistic outcomes in non-relativistic quantum mechanics. We consider a pair of particles that interact in a one-dimensional setting via a delta-function potential. One of the particles is confined to a one-dimensional box, and the other particle is free. The free particle is incident from the left with specified energy, and it may cause changes in state of the confined particle before flying away to the left or to the right. We present a formulation and computational scheme that avoids the use of perturbation theory and determines the probability of any such outcome as a function of the initial state of the confined particle and the energy of the incident particle. As demonstrated by a direct comparison, this presented method holds multiple advantages over a standard perturbative method. The problem formulation and corresponding computational scheme may have applications in physical settings which admit one-dimensional scattering, e.g., in the study of quantum wires or quantum dots.

Figures (9)

• Figure 1: Schematic of scattering problem considered in the present paper. (a) A free incident particle (pink), associated with wavenumber $k_0$, approaches from the left. A trapped particle (blue) within the interval $[0,L]$ (green) is associated with energy index $n_0$. (b) The particles interact via a delta-function potential which has strength $\mu_0$. (c) Possible outcomes involve the incident particle scattering to the left (top) or the right (bottom) with some resultant wavenumber $k$, and leaving the trapped particle in some energy state $n$. (d) The interaction takes place along the line $x_1=x_2$ in configuration space, where $x_1$ and $x_2$ are respectively the positions of the incident and trapped particles.
• Figure 2: Probability structure with $n_0=1$ and a high interaction strength. The parameter $\epsilon$ ranges from $\mathcal{O}(10^3)-\mathcal{O}(10^5)$. (a-c) are associated with repulsive interaction, and (d-f) are associated with attractive interaction. (a,d) represent total probabilities indexed by the outcome $n$; (b,e) represent reflection probabilities indexed by $n$; and (c,f) represent transmission probabilities indexed by $n$.
• Figure 3: Probability structure with $n_0=1$ and a moderate interaction strength. The parameter $\epsilon$ ranges from $\mathcal{O}(0.1)-\mathcal{O}(10)$. (a-c) are associated with repulsive interaction, and (d-f) are associated with attractive interaction. (a,d) represent total probabilities indexed by the outcome $n$; (b,e) represent reflection probabilities indexed by $n$; and (c,f) represent transmission probabilities indexed by $n$.
• Figure 4: Probability structure with $n_0=1$ and a weak relative interaction strength. The parameter $\epsilon$ ranges from $\mathcal{O}(10^{-9})-\mathcal{O}(10^{-7})$. (a-c) are associated with repulsive interaction, and (d-f) are associated with attractive interaction. (a,d) represent total probabilities indexed by the outcome $n$; (b,e) represent reflection probabilities indexed by $n$; and (c,f) represent transmission probabilities indexed by $n$.
• Figure 5: Probability structure with $n_0=5$ and a high interaction strength. The parameter $\epsilon$ ranges from $\mathcal{O}(10^3)-\mathcal{O}(10^5)$. (a-c) are associated with repulsive interaction, and (d-f) are associated with attractive interaction. (a,d) represent total probabilities indexed by the outcome $n$; (b,e) represent reflection probabilities indexed by $n$; and (c,f) represent transmission probabilities indexed by $n$.