Application of Pseudo-Hermitian Quantum Mechanics to a Complex Scattering Potential with Point Interactions

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^2/2m+\zeta_-\delta(x+a)+\zeta_+\delta(x-a), where \zeta_\pm and a are respectively complex and real parameters and \delta(x) is the Dirac delta function. For regions in the space of coupling constants \zeta_\pm where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator \eta and the corresponding equivalent Hermitian Hamiltonian h. \eta turns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, \Im(\zeta_+) = -\Im(\zeta_-). This in particular contains the PT-symmetric case: \zeta_+ = \zeta_-^*. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of \rh or equivalently the non-Hermitian nature of \rH. We show that these physical quantities are not directly sensitive to the presence of PT-symmetry.Comment: 22 pages, 4 figure

Physical Aspects of Pseudo-Hermitian and PTPT-Symmetric Quantum Mechanics

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 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, 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 the conventional quantum mechanics and the classical mechanics.Comment: 45 pages, 13 figures, 2 table

Solvability and PT-symmetry in a double-well model with point interactions

We show that and how point interactions offer one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (usually called PT-symmetric) quantum-mechanical systems. A double-well model is chosen, an easy solvability of which clarifies the mechanisms of the unavoided level crossing and of the spontaneous PT-symmetry breaking. The latter phenomenon takes place at a certain natural boundary of the domain of the "acceptable" parameters of the model. Within this domain the model mediates a nice and compact explicit illustration of the not entirely standard probabilistic interpretation of the physical bound states in the very recently developed (so called PT symmetric or, in an alternative terminology, pseudo-Hermitian) new, fairly exciting and very quickly developing branch of Quantum Mechanics.Comment: 24 p., written for the special journal issue "Singular Interactions in Quantum Mechanics: Solvable Models". Will be also presented to the int. conference "Pseudo-Hermitian Hamiltonians in Quantum Physics III" (Instanbul, Koc University, June 20 - 22, 2005) http://home.ku.edu.tr/~amostafazadeh/workshop/workshop.ht

Coherent and squeezed states of quantum Heisenberg algebras

Starting from deformed quantum Heisenberg Lie algebras some realizations are given in terms of the usual creation and annihilation operators of the standard harmonic oscillator. Then the associated algebra eigenstates are computed and give rise to new classes of deformed coherent and squeezed states. They are parametrized by deformed algebra parameters and suitable redefinitions of them as paragrassmann numbers. Some properties of these deformed states also are analyzed.Comment: 32 pages, 3 figure