We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modeling the propagation of electromagnetic waves traveling in a waveguide half and half filled with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter ζ, we show that the equivalent Hermitian Hamiltonian has the form p2∕2m+(ζ2∕2)∑n=0∞{αn(x),p2n} with αn(x) vanishing outside an interval that is three times larger than the support of v(x), i.e., in 2∕3 of the physical interaction region the potential v(x) vanishes identically. We provide a physical interpretation for this unusual behavior and comment on the classical limit of the system.
We extend the application of the techniques developed within the framework of the pseudo-Hermitian quantum mechanics to study a unitary quantum system described by an imaginary PT-symmetric potential v(x) having a continuous real spectrum. For this potential that has recently been used, in the context of optical potentials, for modeling the propagation of electromagnetic waves traveling in a waveguide half and half filled with gain and absorbing media, we give a perturbative construction of the physical Hilbert space, observables, localized states, and the equivalent Hermitian Hamiltonian. Ignoring terms of order three or higher in the non-Hermiticity parameter ζ, we show that the equivalent Hermitian Hamiltonian has the form p2∕2m+(ζ2∕2)∑n=0∞{αn(x),p2n} with αn(x) vanishing outside an interval that is three times larger than the support of v(x), i.e., in 2∕3 of the physical interaction region the potential v(x) vanishes identically. We provide a physical interpretation for this unusual behavior and comment on the classical limit of the system.
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.
We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian H = p 2 /2 m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h . η turns out to be a (perturbatively) bounded operator for the cases where the imaginary part of the coupling constants have the opposite sign, ℑ(ζ + ) = −ℑ(ζ − ). This in particular contains the -symmetric case: ζ + = ζ* − . We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H . We show that these physical quantities are not directly sensitive to the presence of the -symmetry.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.
A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools, present their utility in establishing a lucid and precise formulation of a unitary quantum theory based on a non-Hermitian Hamiltonian, and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as [Formula: see text], the true meaning and significance of the so-called charge operators [Formula: see text] and the [Formula: see text]-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications and manifestations of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos and biophysics.
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT -symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians H = p 2 + x 2 (i x ) ν with ν (−2, ∞ ) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of H for the cases that ν is an even integer, so that H = p 2 ± x 2+ν , and give a proof of the discreteness of the spectrum of H for all ν (−2, ∞ ). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT -symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT -symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT -symmetric quantum mechanics based on the charge-conjugation operator. Our results provide conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT -symmetric systems regardless of whether they are defined along the real line or a complex contour.
We describe a method that allows for a practical application of the theory of pseudo-Hermitian operators to PT-symmetric systems defined on a complex contour. We apply this method to study the Hamiltonians H=p2+x2(ix)ν with ν∈(−2,∞) that are defined along the corresponding anti-Stokes lines. In particular, we reveal the intrinsic non-Hermiticity of H for the cases that ν is an even integer, so that H=p2±x2+ν, and give a proof of the discreteness of the spectrum of H for all ν∈(−2,∞). Furthermore, we study the consequences of defining a square-well Hamiltonian on a wedge-shaped complex contour. This yields a PT-symmetric system with a finite number of real eigenvalues. We present a comprehensive analysis of this system within the framework of pseudo-Hermitian quantum mechanics. We also outline a direct pseudo-Hermitian treatment of PT-symmetric systems defined on a complex contour which clarifies the underlying mathematical structure of the formulation of PT-symmetric quantum mechanics based on the charge-conjugation operator. Our results provide a conclusive evidence that pseudo-Hermitian quantum mechanics provides a complete description of general PT-symmetric systems regardless of whether they are defined along the real line or a complex contour.