Non-anti-hermitian Quaternionic Quantum Mechanics

Giardino Filho, Sérgio Augusto

BLASI A, G. Scolarici, Luigi Solombrino

Sergio Giardino

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.

Lawrence Paul Horwitz, Lawrence C. Biedenharn, Daniel Sepunaru

Alberto Blasi, G. Scolarici, Luigi Solombrino

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.

František Jenč