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Experiments in PT −symmetric quantum mechanics

Miloslav Znojil

Published Jan 1, 2003• 20 citations
OpenAlex

Abstract

Extended quantum mechanics using non-Hermitian, pseudo-Hermitian Hamiltonians is briefly reviewed. Supersymmetric regularizations, solvable simulations and large-N expansion techniques are recollected as suitable means for the study of non-local quasi-particles. Their many-particle or even, perhaps, unstable and decaying generalizations are also considered. Picking up the Klein-Gordon equation and its Feshbach-Villars' Hamiltonian for definiteness, we argue that the PT symmetry might help to clarify experimental aspects of relativistic quantum mechanics.

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We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.

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For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.

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The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.

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Should PT symmetric quantum mechanics be interpreted as nonlinear?

Miloslav Znojil

PT symmetric quantum mechanics admits non-Hermitian Hamiltonians with real spectra but, as long as the evolution is pseudo-unitary, the probabilistic interpretation of wave functions is lost. We offer an effective Hermitization of the theory using the Feshbach-type elimination of ghosts. The energy-dependence of the resulting Hermitian effective Hamiltonians Heff is a nonlinearity which, in a strong-coupling regime, explains the spontaneous PT symmetry breaking and the simultaneous loss of the Hermiticity of Heff while the pseudo-unitary evolution law persists. PACS 03.65.Bz, 03.65.Ca 03.65.Fd The concept of extended, non-Hermitian quantum mechanics [1] with real spectra and PT symmetric Hamiltonians (P means parity and T denotes time reversal) grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [2] have discovered that a

•Jan 1, 2001•20 citations
Extended quantum mechanics using non-Hermitian, pseudo-Hermitian Hamiltonians is briefly reviewed. Supersymmetric regularizations, solvable simulations and large-N expansion techniques are recollected as suitable means for the study of non-local quasi-particles. Their many-particle or even, perhaps, unstable and decaying generalizations are also considered. Picking up the Klein-Gordon equation and its Feshbach-Villars' Hamiltonian for definiteness, we argue that the PT symmetry might help to clarify experimental aspects of relativistic quantum mechanics.
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-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of 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 symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a canonical orthonormal basis in which a previously introduced unitary mapping of H to a Hermitian Hamiltonian h takes a simple form. We use this basis to construct the observables O α of the quantum mechanics based on H . In particular, we introduce pseudo-Hermitian position and momentum operators and a pseudo-Hermitian quantization scheme that relates the latter to the ordinary classical position and momentum observables. These allow us to address the problem of determining the conserved probability density and the underlying classical system for pseudo-Hermitian and in particular PT -symmetric quantum systems. As a concrete example we construct the Hermitian Hamiltonian h , the physical observables O α , the localized states and the conserved probability density for the non-Hermitian PT -symmetric square well. We achieve this by employing an appropriate perturbation scheme. For this system, we conduct a comprehensive study of both the kinematical and dynamical effects of the non-Hermiticity of the Hamiltonian on various physical quantities. In particular, we show that these effects are quantum mechanical in nature and diminish in the classical limit. Our results provide an objective assessment of the physical aspects of PT -symmetric quantum mechanics and clarify its relationship with both conventional quantum mechanics and classical mechanics.
The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.
PT symmetric quantum mechanics admits non-Hermitian Hamiltonians with real spectra but, as long as the evolution is pseudo-unitary, the probabilistic interpretation of wave functions is lost. We offer an effective Hermitization of the theory using the Feshbach-type elimination of ghosts. The energy-dependence of the resulting Hermitian effective Hamiltonians Heff is a nonlinearity which, in a strong-coupling regime, explains the spontaneous PT symmetry breaking and the simultaneous loss of the Hermiticity of Heff while the pseudo-unitary evolution law persists. PACS 03.65.Bz, 03.65.Ca 03.65.Fd The concept of extended, non-Hermitian quantum mechanics [1] with real spectra and PT symmetric Hamiltonians (P means parity and T denotes time reversal) grew from several sources. The oldest root of its appeal is the Rayleigh-Schrödinger perturbation theory. Within its framework, Caliceti et al [2] have discovered that a