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Non-anti-hermitian Quaternionic Quantum Mechanics

Sergio Giardino

Published Jan 1, 2018• 20 citations
OpenAlex

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-anti-hermitian 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.

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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-anti-hermitian 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.
We characterize the quasianti-Hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra; moreover, we state a necessary and sufficient condition for a set of quasianti-Hermitian quaternionic operators to be anti-Hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional (right) quaternionic Hilbert space. According to such results we obtain two alternative descriptions of a quantum optical physical system, in the realm of quaternionic quantum mechanics, while no alternative can exist in complex quantum mechanics, and we discuss some differences between them.
The quantum evolution of the Wigner function for Gaussian wave packets generated by a non-Hermitian Hamiltonian is investigated. In the semiclassical limit $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ this yields the non-Hermitian analog of the Ehrenfest theorem for the dynamics of observable expectation values. The lack of Hermiticity reveals the importance of the complex structure on the classical phase space: The resulting equations of motion are coupled to an equation of motion for the phase-space metric---a phenomenon having no analog in Hermitian theories.
ψ-function, the Generalized Ehrenfest Theorem re- veals some interesting exact relationships between quantum and classical expectation values. These general results seem to imply a deep-rooted unity (in spite of apparent radical differences in conceptual structures) between classical and quantum mechan- ics. Some significant consequences and important physical insights which follow from the general formulation are discussed with examples. Most im- portant is that it offers, under reasonable approx i- mations, a pure quantum mechanical description of the Stern-Gerlach experiment with realistic inhomo- geneous magnetic field (∇ ∇.B = 0). EVEN almost seventy-five years after the first formul a- tion of quantum mechanics (QM), its relation with clas - sical mechanics (CM) remains obscure. The usual belief that QM has superseded CM is also not well founded. In all cases where a new theory supersedes an old one, there is a universal prescription of recovering the mathematical structure of the old theory in the doma in of its validity, i.e. where it is empirically established. This is true for both special and general theory of rel a- tivity 1 . But for QM, no such universal approximation procedure can be offered. About the success and failure of the two mechanics we can make two positive asser- tions: (i) There are experimental results in the micr o- scopic domain (such as hydr ogen spectrum) or in the macroscopic domain (such as specific heat of so lids at low temperature) which cannot be interpreted within the framework of CM. (ii) In all such cases, the mathemat i- cal formalism of QM provides the correct empirical results. Not a single failure has yet been reported. These empirical evidences show that CM cannot be the universal mechanics and that QM has such a pote n- tiality. However, since it is not yet possible to recover CM in all the domains of its validity, we cannot assert that QM has superseded CM. A second alternative that