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Ali Mostafazadeh

Publications and source records attributed to Ali Mostafazadeh.

At least 109 records · Page 6Linked to original sources
arXiv2010

Spectral Singularities, Biorthonormal Systems, and a Two-Parameter Family of Complex Point Interactions

A curious feature of complex scattering potentials v(x) is the appearance of spectral singularities. We offer a quantitative description of spectral singularities that identifies them with an obstruction to the existence of a complete biorthonormal system consisting of the eigenfunctions of the Hamiltonian operator, i.e., -\frac{d^2}{dx^2}+v(x), and its adjoint. We establish the equivalence of this description with the mathematicians' definition of spectral singularities for the potential v(x)=z_-δ(x+a)+z_+δ(x-a), where z_\pm and a are respectively complex and real parameters and δ(x) is the Dirac delta-function. We offer a through analysis of the spectral properties of this potential and determine the regions in the space of the coupling constants z_\pm where it admits bound states and spectral singularities. In particular, we find an explicit bound on the size of certain regions in which the Hamiltonian is quasi-Hermitian and examine the consequences of imposing PT-symmetry.

math-ph↗
arXiv2010

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+ζ_-δ(x+a)+ζ_+δ(x-a), where ζ_\pm and a are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ_\pm where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator ηand the corresponding equivalent Hermitian Hamiltonian h. ηturns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, \Im(ζ_+) = -\Im(ζ_-). This in particular contains the PT-symmetric case: ζ_+ = ζ_-^*. 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.

quant-ph↗
arXiv2010

Resonance Phenomenon Related to Spectral Singularities, Complex Barrier Potential, and Resonating Waveguides

A peculiar property of complex scattering potentials is the appearance of spectral singularities. These are energy eigenvalues for certain scattering states that similarly to resonance states have infinite reflection and transmission coefficients. This property reveals an interesting resonance effect with possible applications in waveguide physics. We study the spectral singularities of a complex barrier potential and explore their application in designing a waveguide that functions as a resonator. We show that for the easily accessible sizes of the waveguide and its gain region, we can realize the spectral singularity-related resonance phenomenon at almost every wavelength within the visible spectrum or outside it.

quant-ph↗
arXiv2010

Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at real Energies

Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a wave guide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic wave guide.

math-ph↗
arXiv2010

Pseudo-Hermiticity and Electromagnetic Wave Propagation in Dispersive Media

Pseudo-Hermitian operators appear in the solution of Maxwell's equations for stationary non-dispersive media with arbitrary (space-dependent) permittivity and permeability tensors. We offer an extension of the results in this direction to certain stationary dispersive media. In particular, we use the WKB approximation to derive an explicit expression for the planar time-harmonic solutions of Maxwell's equations in an inhomogeneous dispersive medium and study the combined affect of inhomogeneity and dispersion.

quant-ph↗
arXiv2009

Quantum Mechanics of Proca Fields

We construct the most general physically admissible positive-definite inner product on the space of Proca fields. Up to a trivial scaling this defines a five-parameter family of Lorentz invariant inner products that we use to construct a genuine Hilbert space for the quantum mechanics of Proca fields. If we identify the generator of time-translations with the Hamiltonian, we obtain a unitary quantum system that describes first-quantized Proca fields and does not involve the conventional restriction to the positive-frequency fields. We provide a rather comprehensive analysis of this system. In particular, we examine the conserved current density responsible for the conservation of the probabilities, explore the global gauge symmetry underlying the conservation of the probabilities, obtain a probability current density, construct position, momentum, helicity, spin, and angular momentum operators, and determine the localized Proca fields. We also compute the generalized parity ($\cP$), generalized time-reversal ($\cT$), and generalized charge or chirality ($\cC$) operators for this system and offer a physical interpretation for its $\cP\cT$-, $\cC$-, and $\cC\cP\cT$-symmetries.

quant-ph↗
arXiv2009

On Hamiltonians Generating Optimal-Speed Evolutions

We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric- (inner product-) dependence of the lower bound on the travel time and the universality (metric-independence) of the upper bound on the speed of unitary evolutions.

quant-ph↗
arXiv2008

Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.

quant-ph↗
arXiv2008

QT-Symmetry and Weak Pseudo-Hermiticity

For an invertible (bounded) linear operator Q acting in a Hilbert space ${\cal H}$, we consider the consequences of the QT-symmetry of a non-Hermitian Hamiltonian $H:{\cal H}\to{\cal H}$ where T is the time-reversal operator. If H is symmetric in the sense that ${\cal T} H^\dagger {\cal T}=H$, then QT-symmetry is equivalent to Q^{-1}-weak-pseudo-Hermiticity. But in general this equivalence does not hold. We show this using some specific examples. Among these is a large class of non-PT-symmetric Hamiltonians that share the spectral properties of PT-symmetric Hamiltonians.

quant-ph↗
arXiv2008

Propagation of Electromagnetic Waves in Linear Media and Pseudo-Hermiticity

We express the electromagnetic field propagating in an arbitrary time-independent non-dispersive medium in terms of an operator that turns out to be pseudo-Hermitian for Hermitian dielectric and magnetic permeability tensors Epsilon and Mu. We exploit this property to determine the propagating field. In particular, we obtain an explicit expression for a planar field in an isotropic medium with Epsilon and Mu varying along the direction of the propagation. We also study the scattering of plane waves due to a localized inhomogeneity.

physics.optics↗
arXiv2007

Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.

quant-ph↗
arXiv2007

Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''

In arXiv:0710.5653v1 M. Znojil claims that he has found and corrected an error in my paper: [Phys. Lett. B \textbf{650}, 208 (2007), arXiv:0706.1872v2] and that it is possible to escape its main conclusion, namely that the unitarity of the time-evolution and observability of the Hamiltonian imply time-independence of the metric operator. In this note I give a very short calculation showing that the analysis given by M. Znojil also leads to the same conclusion as of my above-mentioned paper. This is actually a reconfirmation of the validity of the results of the latter paper.

quant-ph↗
arXiv2007

Time-Dependent Pseudo-Hermitian Hamiltonians Defining a Unitary Quantum System and Uniqueness of the Metric Operator

The quantum measurement axiom dictates that physical observables and in particular the Hamiltonian must be diagonalizable and have a real spectrum. For a time-independent Hamiltonian (with a discrete spectrum) these conditions ensure the existence of a positive-definite inner product that renders the Hamiltonian self-adjoint. Unlike for a time-independent Hamiltonian, this does not imply the unitarity of the Schroedinger time-evolution for a general time-dependent Hamiltonian. We give an additional necessary and sufficient condition for the unitarity of time-evolution. In particular, we obtain the general form of a two-level Hamiltonian that fulfils this condition. We show that this condition is geometrical in nature and that it implies the reality of the adiabatic geometric phases. We also address the problem of the uniqueness of the metric operator.

quant-ph↗
arXiv2007

Physical Meaning of Hermiticity and Shortcomings of the Composite (Hermitian + non-Hermitian) Quantum Theory of Gunther and Samsonov

In arXiv:0709.0483 Gunther and Samsonov outline a ``generalization'' of quantum mechanics that involves simultaneous consideration of Hermitian and non-Hermitian operators and promises to be ``capable to produce effects beyond those of standard Hermitian quantum mechanics.'' We give a simple physical interpretation of Hermiticity and discuss in detail the shortcomings of the above-mentioned composite quantum theory. In particular, we show that the corresponding ``generalization of measurement theory'' suffers from a dynamical inconsistency and that it is by no means adequate to replace the standard measurement theory.

quant-ph↗
arXiv2006

Explicit Realization of Pseudo-Hermitian and Quasi-Hermitian Quantum Mechanics for Two-Level Systems

We give an explicit characterization of the most general quasi-Hermitian operator H, the associated metric operators η_+, and η_+-pseudo-Hermitian operators acting in two-dimensional complex Euclidean space C^2. These operators represent the physical observables of a model whose Hamiltonian and Hilbert space are respectively H and C^2 endowed with the inner product defined by η_+. Our calculations allows for a direct demonstration of the fact that the choice of an irreducible family of observables fixes the metric operator up to a multiplicative factor.

quant-ph↗