Quaternionic elastic scattering

Abstract

We study the elastic scattering of quantum particles based on a real Hilbert space approach to quaternionic quantum mechanics ( QM) and derive expression for the wave function, the phase shifts, as well as the optical theorem for the case of a hard sphere scattering potential. The strong agreement between these new quaternionic results and the corresponding results in complex quantum mechanics reinforce the validity of the QM generalization of ordinary complex quantum mechanics ( QM).

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 scalar field in the real Hilbert space

“…This quantization method is simpler than the previous one, because Θ = Θ 0 . Using the symplectic notation (4), the complex equations of motion (21) can be obtained from the lagrangian density…”

“…In recent past, a variant of ÀQM has been developed over the real Hilbert space [15,16], and the hope for a coherent quaternionic quantum theory has been renewed. The real Hilbert space approach to ÀQM restores Ehrenfest's theorem and the classical limit, and simple quaternionic quantum solutions that were never obtained in the antihermitian theory have been achieved [17,18,19,20,21,22,23]. The ÀQM has also been extended to the relativistic theory, and quaternionic versions of the Klein-Gordon [24], and of the Dirac equations [25] have been successfully obtained.…”

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].…”