Non-Stationary Non-Hermitian "Wrong-Sign" Quantum Oscillators and Their Meaningful Physical Interpretation

• Published in: Entropy (Basel, Switzerland) Switzerland), 2023-04, Vol.25 (4), p.692
• Main Author: Znojil, Miloslav
• Format: Article
• Language: English
• Subjects: a complete set of evolution equations; application to asymptotically repulsive potential models; Generators; Hilbert space; internal consistency of the probabilistic interpretation; non-Hermitian interaction representation; Operators; Oscillators; Quantum mechanics; quantum mechanics of non-stationary unitary systems; Quantum physics; Quantum theory; quasi-Hermitian form of the observables; Textbooks
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e25040692

In the framework of quantum mechanics using quasi-Hermitian operators the standard unitary evolution of a non-stationary but still closed quantum system is only properly described in the non-Hermitian interaction picture (NIP). In this formulation of the theory both the states and the observables vary with time. A few aspects of implementation of this picture are illustrated via the "wrong-sign" quartic oscillators. It is shown that in contrast to the widespread belief, both of the related Schrödinger-equation generators G(t) and the Heisenberg-equation generators Σ(t) are just auxiliary concepts. Their spectra are phenomenologically irrelevant and, in general, complex. It is argued that only the sum H(t)=G(t)+Σ(t) of the latter operators retains the standard physical meaning of the instantaneous energy of the unitary quantum system in question.