Complex quantum momentum due to correlation

TL;DR

This paper investigates how attractive interactions between electrons in a one-dimensional quantum well necessitate complex momentum solutions, highlighting the fundamental role of complex numbers in quantum mechanics.

Contribution

It demonstrates that attractive electron interactions lead to the emergence of complex momentum solutions, emphasizing the importance of complex numbers in quantum theory.

Findings

Complex momentum solutions arise from attractive interactions.

Perturbative analysis and Newton's method are used to solve complex momenta.

Probability densities from complex solutions differ from real momentum cases.

Abstract

Real numbers provide a sufficient description of classical physics and all measurable phenomena; however, complex numbers are occasionally utilized as a convenient mathematical tool to aid our calculations. On the other hand, the formalism of quantum mechanics integrates complex numbers within its fundamental principles, and whether this arises out of necessity or not is an important question that many have attempted to answer. Here, we will consider two electrons in a one-dimensional quantum well where the interaction potential between the two electrons is attractive as opposed to the usual repulsive coulomb potential. Pairs of electrons exhibiting such effective attraction towards each other occur in other settings, namely within superconductivity. We will demonstrate that this attractive interaction leads to the necessity of complex momentum solutions, which further emphasizes the significance of complex numbers in quantum theory. The complex momentum solutions are solved using a perturbative analysis approach in tandem with Newton's method. The probability densities arising from these complex momentum solutions allow for a comparison with the probability densities of the typical real momentum solutions occurring from the standard repulsive interaction potential.