Quaternionic Quantum Particles

Abstract

If Ψ is a quaternionic wave function, then iΨ = Ψi. Thus, there are two versions of the quaternionic Schrödinger equation (QSE). In this article, we present the second possibility for solving the QSE, following on from a previous article. After developing the general methodology, we present the quaternionic free particle solution and the scattering of the quaternionic particle through a scalar barrier.

Quaternionic Dirac free particle

“…where Φ * and Ψ * are quaternionic conjugates. This real inner product establishes the quantum expectation value in the real Hilbert space ÀQM, and the consistency demonstrated in the non-relativistic results [11,12,13,10,14,15,16] also encourages us to apply it the real Hilbert space ÀQM formalism to every quantum system. Let us then consider the relativistic quantum problem we want to study.…”

“…In spite of these difficulties, further efforts are currently being carried out to find alternative formulations to ÀQM, and we quote [6,7] by way of example. However, the most consistent impulse to ÀQM in recent times is the development of the real Hilbert space approach [8,9], where the Ehrenfest theorem has been proven, and a diversified collection of results has been obtained [10,11,12,13,14,15,16]. These outcomes within the non-relativistic theory suggested the hypothesis of a relativistic ÀQM, whose first achievement was established by the solution of the quaternionic Klein-Gordon equation (ÀKGE) [17].…”

Quaternionic quantum harmonic oscillator

“…Section 4.4 of [6]) is the physical motivation to the introduction of the real Hilbert space formalism to ÀQM. The consistency demonstrated in these previous results [34][35][36][37][38][39][40] encourage us to apply the real Hilbert space ÀQM formalism to quantum systems that do not have satisfactory quaternionic interpretations.…”

“…More recently, a novel approach eliminated the anti-hermiticity requirement for the Hamiltonian operator in ÀQM [32,33]. Using this framework, several results have been obtained, including the explicit solutions of the Aharonov-Bohm effect [34], the free particle [35,36], the square well [37], the Lorentz force [38,39] and the quantum scattering [40,41]. Further conceptual results are the well defined classical limit [32], the virial theorem [38], the Ehrenfest theorem and the real Hilbert space [32,33].…”

Quaternionic Klein-Gordon equation

“…Using this approach, the anti-hermitian requirement of the Hamiltonian operator was removed, and a simpler theory emerged. The consistency and simplicity of the real Hilbert space approach enabled us to elucidate several unsolved problems of ÀQM, specifically the Aharonov-Bohm effect [15], the free particle [16,17], the square well [18], the Lorentz force [13,19], the quantum scattering [20,21], and the harmonic oscillator [22].…”