  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.
