Self-interacting quantum particles and the Dirac delta potential

TL;DR

This paper investigates self-interacting quantum particles in the Dirac delta potential within a real Hilbert-space formulation that unifies complex and quaternionic wave functions. By first solving the Dirac delta problem in the complex case, the work establishes a mathematically general approach and reveals non-stationary, non-conservative features that arise from the real-Hilbert-space definition of observables. It then extends the framework to quaternionic quantum mechanics, where the pure imaginary quaternionic component of the potential induces a self-interaction between the complex and quaternionic parts, yielding richer bound and scattering structures and modifying momentum. The results show that complex quantum mechanics emerges as a limit of the quaternionic real-Hilbert-space theory, while quaternionic self-interaction provides new phenomena absent in $ ext{C}$QM, supporting the broader viability and potential experimental relevance of $ ext{H}$QM in this real setting.

Abstract

The Dirac delta function potential is considered within the real Hilbert space approach for complex wave functions, as well as quaternionic wave functions. As has been previously determined, the real Hilbert space approach enables the possibility of self-interacting physical systems. The self-interaction precludes confining states, and also imposes non-stationary quantum states, both of them representing novel situations that cannot be observed in terms of quantum wave functions. These results remark the differences between quaternionic quantum mechanics ($\mathbbm H$QM) and complex quantum mechanics ($\mathbbm C$QM), and also establish a method of solving the wave equation that may be applied to a variety of different cases.