Quasi-Hermitian operators in quantum mechanics and the variational principle

We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. This involves the construction of a metric (if it exists) for the given set of non-Hermitian observables. We disc...

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Bibliographic Details
Published inAnnals of physics Vol. 213; no. 1; pp. 74 - 101
Main Authors Scholtz, F.G., Geyer, H.B., Hahne, F.J.W.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 1992
Elsevier
Subjects
Classical and quantum physics: mechanics and fields
Exact sciences and technology
Physics
Quantum mechanics
Ground state
Metric
Quantum mechanics
Mathematical operator
Variational principle
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Summary:We establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation. This involves the construction of a metric (if it exists) for the given set of non-Hermitian observables. We discuss uniqueness of this metric. We also show that it is not always necessary to construct the metric for the whole set of observables under consideration, but that it is sufficient for some calculational purposes to construct it for a subset only, even though this metric is, in general, not unique. The restricted metric turns out to be particularly useful in the implementation of a variational principle, which we also formulate.
ISSN:0003-4916
1096-035X
DOI:10.1016/0003-4916(92)90284-S