Non-anti-hermitian Quaternionic Quantum Mechanics

Abstract

The breakdown of Ehrenfest's theorem imposes serious limitations on quaternionic quantum mechanics (QQM). In order to determine the conditions in which the theorem is valid, we examined the conservation of the probability density, the expectation value and the classical limit for a non-antihermitian formulation of QQM. The results also indicated that the non-anti-hermitian quaternionic theory is related to non-hermitian quantum mechanics, and thus the physical problems described with both of the theories should be related.

Virial theorem and generalized momentum in quaternionic quantum mechanics

“…In this article we have revisited and emended the mathematical machinery of HQM in real Hilbert space [30][31][32]. The previous results demonstrate that the theory is provided with wave equation, momentum operator, conservation of probability, expectation values, classical limit and spectral decomposition.…”

“…A second element of the non-anti-hermitian HQM comes by analogy with CQM, where the momentum expectation value is related to the probability current through Π = J /m. This analogy has been used in [31] to define the expectation value for an arbitrary quaternic operator O as…”

Quaternionic Quantum Particles

“…A more radical approach emerged after the discovery of quaternionic solutions obtained throughout non-anti-hermitian (NAH) Hamiltonians in the study of the quaternionic Aharonov-Bohm (AB) effect [21]. This discovery has enabled a formal expression of an NAH-QQM, where the probability current and the expectation value are redefined [22]. Following these results, a solution for the QSE was developed, and the first quaternionic particle solution was obtained [23].…”

New scattering features of quaternionic point interaction in non-Hermitian quantum mechanics

“…Ref [18] posed the question on the necessity of anti-Hermitian assumption in QQM. Ref [19] propose a non-anti-Hermitian QQM as quaternionic generalization of previously known (non-quaternionic) non-Hermitian quantum mechanics (NHQM). The non-Hermitian systems with real energy eigenvalues has become the topic of frontier research over the last two decades.…”

“…These are also called as zero width resonance and found to be extremely sensitive to the dimension of interacting region [55]. As stated above it is proposed that [19] non-anti-Hermitian QQM can be possibly formulated as a generalization of NHQM, it is most desirable to study the scattering features in the domain of non-anti-Hermitian QQM and study for the possible deviation in the scattering features between non-anti-Hermitian QQM and NHQM. To provide analytical understanding (and thus avoiding transcendental equations), we chose point interaction represented by quaternionic (single) delta potential for our study of spectral singularity in non-anti-Hermitian QQM.…”