Quantum self-interaction within an infinitely deep cavity

TL;DR

This work analyzes the infinitely deep quantum well within a real Hilbert space framework of quaternionic quantum mechanics, demonstrating that self-interaction, absent in standard complex QM, naturally arises in quaternionic formulations. It first reproduces the complex case results, then presents a totally general solution for the quaternionic infinite well, revealing a modified energy spectrum E1(N) = sqrt{E_N^2 + |U1|^2} and a richer set of stationary and non-stationary states. The study also shows how self-interaction via the quaternionic component U1 affects spectral structure while leaving position and momentum observables largely aligned with complex results, and discusses alternative quaternionic formulations that influence temporal versus spatial stationarity. Overall, the paper supports the quaternionic real-Hilbert-space framework as a viable extension of quantum theory, offering novel spectral features and directions for future one-dimensional and relativistic quantum investigations.

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.