Quantum self-interaction within an infinitely deep cavity

TL;DR

This paper explores the quantum infinite square well in both complex and quaternionic frameworks, revealing new self-interaction phenomena and extending the understanding of quantum solutions beyond traditional complex Hilbert space.

Contribution

It introduces quaternionic wave functions into the quantum infinite well, uncovering self-interaction effects not observable in standard complex solutions.

Findings

Complex solutions reproduce known results and extend to non-stationary states.

Quaternionic solutions exhibit self-interaction effects.

Both frameworks broaden the scope of one-dimensional quantum solutions.

Abstract

One examines the infinitely deep quantum cavity, also known as the quantum infinite square well, within the framework of the real Hilbert space. The solutions are considered in terms of complex wave functions, and also in terms of quaternionic wave functions. The complex results reproduce the usual achievements established in the complex Hilbert space, but also extend them to non-stationary solutions, as well as to distorted stationary solutions, different energy spectra, and dislocated observed position. The quaternionic cases further admit the incidence of self-interaction, something that cannot be observed in complex solutions. Therefore, both the complex and quaternionic solutions are more general than previous cases, thus opening the way to further one-dimensional solutions to be researched in the non-relativistic theory.