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Zeitschrift für Naturforschung A

Local States in One-Dimensional Symmetrical Quantum Systems

Jürgen Brickmann

Published Dec 1, 1979• 1 citation
Publisher

Abstract

Abstract Local quantum states, which play an important role in quantum dynamical treatments, are expanded analytically with respect to a basis of eigen functions of a symmetrical Hamiltonian ℋ̂(x) = ℋ̂(- x). Exact local states (ELS) in one-dimensional symmetrical quantum systems are therein defined as quantum states which are local eigenstates of the Hamiltonian ℋ̂(x) on one half space ℝ+ or ℝ- and are identically equal to zero on the other half space. Local properties like the projection operator on one half space can be given in terms of ELS-basis, but it is shown that the energy moments 〈(〈ℋ̂ 〉 - 〈ℋ̂)k〉 with respect to the ELS do not converge. Consequently, if one uses the ELS as quasistationary initial states, as has been done recently by some authors [5], the lifetimes of these states cannot be estimated from time energy uncertainty relation using the second energy moment as an energy uncertainty measure. A harmonic oscillator system and a symmetrical double oscillator are treated as examples.

DOI
10.1515/zna-1979-1209

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Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described.
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It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT\mathcal{PT}PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT\mathcal{PT}PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT\mathcal{PT}PT symmetry have been applied in the context of many different branches of physics. This review introduces the concepts of PT\mathcal{PT}PT symmetry by focusing on elementary one-dimensional PT\mathcal{PT}PT-symmetric quantum and classical mechanics and relies, in particular, on oscillator models to illustrate and explain the basic properties of PT\mathcal{PT}PT-symmetric quantum theory.