Showing papers on "Ladder operator published in 2009"

A non-Hermitian Hamilton operator and the physics of open quantum systems

Abstract: The Hamiltonian Heff of an open quantum system consists formally of a first-order interaction term describing the closed (isolated) system with discrete states and a second-order term caused by the interaction of the discrete states via the common continuum of scattering states. Under certain conditions, the last term may be dominant. Due to this term, Heff is non-Hermitian. Using the Feshbach projection operator formalism, the solution ΨEc of the Schrodinger equation in the whole function space (with discrete as well as scattering states, and the Hermitian Hamilton operator H) can be represented in the interior of the localized part of the system in the set of eigenfunctions λ of Heff. Hence, the characteristics of the eigenvalues and eigenfunctions of the non-Hermitian operator Heff are contained in observable quantities. Controlling the characteristics by means of external parameters, quantum systems can be manipulated. This holds, in particular, for small quantum systems coupled to a small number of channels. The paper consists of three parts. In the first part, the eigenvalues and eigenfunctions of non-Hermitian operators are considered. Most important are the true and avoided crossings of the eigenvalue trajectories. In approaching them, the phases of the λ lose their rigidity and the values of observables may be enhanced. Here the second-order term of Heff determines decisively the dynamics of the system. The time evolution operator is related to the non-Hermiticity of Heff. In the second part of the paper, the solution ΨEc and the S matrix are derived by using the Feshbach projection operator formalism. The regime of overlapping resonances is characterized by non-rigid phases of the ΨEc (expressed quantitatively by the phase rigidity ρ). They determine the internal impurity of an open quantum system. Here, level repulsion passes into width bifurcation (resonance trapping): a dynamical phase transition takes place which is caused by the feedback between environment and system. In the third part, the internal impurity of open quantum systems is considered by means of concrete examples. Bound states in the continuum appearing at certain parameter values can be used in order to stabilize open quantum systems. Of special interest are the consequences of the non-rigidity of the phases of λ not only for the problem of dephasing, but also for the dynamical phase transitions and questions related to them such as phase lapses and enhancement of observables.

The spectrum of the 1-laplace operator

Abstract: The eigenfunction of the 1-Laplace operator is defined to be a critical point in the sense of the strong slope for a nonsmooth constraint variational problem. We completely write down all these eigenfunctions for the 1-Laplace operator on intervals.

Pseudo-Boson Coherent and Fock States

Abstract: Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

Abstract: We consider a periodic self-adjoint pseudo-differential operator $H=(-\Delta)^m+B$, $m>0$, in $\R^d$ which satisfies the following conditions: (i) the symbol of $B$ is smooth in $\bx$, and (ii) the perturbation $B$ has order less than $2m$. Under these assumptions, we prove that the spectrum of $H$ contains a half-line. This, in particular implies the Bethe-Sommerfeld Conjecture for the Schr\"odinger operator with a periodic magnetic potential in all dimensions.

S-duality, 't Hooft operators and the operator product expansion

Abstract: We study S-duality in = 4 super Yang-Mills with an arbitrary gauge group by determining the operator product expansion of the circular BPS Wilson and 't Hooft loop operators. The coefficients in the expansion of an 't Hooft loop operator for chiral primary operators and the stress-energy tensor are calculated in perturbation theory using the quantum path-integral definition of the 't Hooft operator recently proposed. The corresponding operator product coefficients for the dual Wilson loop operator are determined in the strong coupling expansion. The results for the 't Hooft operator in the weak coupling expansion exactly reproduce those for the dual Wilson loop operator in the strong coupling expansion, thereby demonstrating the quantitative prediction of S-duality for these observables.

The harmonic oscillator in quantum mechanics: A third way

Abstract: Courses on undergraduate quantum mechanics usually focus on solutions of the Schrodinger equation for several simple one-dimensional examples. When the notion of a Hilbert space is introduced, only academic examples are used, such as the matrix representation of Dirac’s raising and lowering operators or the angular momentum operators. We introduce some of the same one-dimensional examples as matrix diagonalization problems, with a basis that consists of the infinite set of square well eigenfunctions. Undergraduate students are well equipped to handle such problems in familiar contexts. We pay special attention to the one-dimensional harmonic oscillator. This paper should equip students to obtain the low lying bound states of any one-dimensional short range potential.

Multipole representation of the Fermi operator with application to the electronic structure analysis of metallic systems

Abstract: The Fermi operator, i.e., the Fermi-Dirac function of the system Hamiltonian, is a fundamental quantity in the quantum mechanics of many-electron systems and is ubiquitous in condensed-matter physics. In the last decade the development of accurate and numerically efficient representations of the Fermi operator has attracted lot of attention in the quest for linear scaling electronic structure methods based on effective one-electron Hamiltonians. These approaches have numerical cost that scales linearly with N, the number of electrons, and thus hold the promise of making quantummechanical calculations of large systems feasible. Achieving linear scaling in realistic calculations is very challenging. Formulations based on the Fermi operator are appealing because this operator gives directly the single-particle density matrix without the need for Hamiltonian diagonalization. At finite temperature the density matrix can be expanded in terms of finite powers of the Hamiltonian, requiring computations that scale linearly with N owing to the sparse character of the Hamiltonian matrix. 1 These properties of the Fermi operator are valid not only for insulators but also for metals, making formulations based on the Fermi operator particularly attractive. Electronic structure algorithms using a Fermi operator expansion FOE were introduced by Baroni and Giannozzi 2 and Goedecker et al. 3,4 see also the review article 5 . These authors proposed polynomial and rational approximations of the Fermi operator. Major improvements were made recently in a series of publications by Parrinello and coauthors, 6‐11 in which a new form of Fermi operator expansion was introduced based on the grand canonical formalism.

Principles of Quantum Mechanics: As Applied to Chemistry and Chemical Physics

Abstract: This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The first two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to Schrodinger's wave mechanics, the postulates of quantum mechanics are presented along with essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics and materials science.

Quantum Inequalities from Operator Product Expansions

Abstract: Quantum inequalities are lower bounds for local averages of quantum observables that have positive classical counterparts, such as the energy density or the Wick square. We establish such inequalities in general (possibly interacting) quantum field theories on Minkowski space, using nonperturbative techniques. Our main tool is a rigorous version of the operator product expansion.

Alternative descriptions and bipartite compound quantum systems

Abstract: We analyse some features of alternative Hermitian and quasi-Hermitian quantum descriptions of simple and bipartite compound systems. We show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as a tensor product of positive operators on component spaces. Some examples also show that such property could be strictly connected with symmetry properties of the non-Hermitian Hamiltonian.

A quantum parametric oscillator in a radiofrequency trap

Abstract: An ion confined within a Paul trap can be assimilated, in good approximation, with a harmonic oscillator. A method which enables the construction of an invariant operator based on the Lewis and Riesenfeld approach is reported. The method is then applied to the case of an ion confined in a Paul trap, treated as a quantum harmonic oscillator. An invariant operator is associated with the system. The spectrum of the quasienergy operator finally results. The eigenvectors of the system are found using the Fock state basis.

Constraint-induced mean curvature dependence of Cartesian momentum operators

Abstract: The Hermitian Cartesian quantum momentum operator $\mathbf{p}$ for an embedded surface $M$ in $R^{3}$ is proved to be a constant factor $-i\hbar $ times the mean curvature vector field $H\mathbf{n}$ added to the usual differential term. With use of this form of momentum operators, the operator-ordering ambiguity exists in the construction of the correct kinetic energy operator and three different operator-orderings lead to the same result. PACS: 03.65.-w Quantum mechanics, 04.60.Ds Canonical quantization

New nonlinear coherent states associated to inverse bosonic and $f$-deformed ladder operators

Abstract: Using the {\it nonlinear coherent states method}, a formalism for the construction of the coherent states associated to {\it "inverse bosonic operators"} and their dual family has been proposed. Generalizing the approach, the "inverse of $f$-deformed ladder operators" corresponding to the nonlinear coherent states in the context of quantum optics and the associated coherent states have been introduced. Finally, after applying the proposal to a few known physical systems, particular nonclassical features as sub-Poissonian statistics and the squeezing of the quadratures of the radiation field corresponding to the introduced states have been investigated.

Coherent states approach to Penning trap

Abstract: By using a matrix technique, which allows us to identify directly the ladder operators, the Penning trap coherent states are derived as eigenstates of the appropriate annihilation operators. These states are compared with those obtained through the displacement operator. The associated wavefunctions and mean values for some relevant operators in these states are also evaluated. It turns out that the Penning trap coherent states minimize the Heisenberg uncertainty relation.

Quantum stochastic integral representations of Fock space operators

Abstract: An (unbounded) operator Ξ on Boson Fock space over L 2(R +) is called regular if it is an admissible white noise operator such that the conditional expectations give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation.

Generalized hypergeometric photon-added and photon-depleted coherent states

Abstract: In this paper, we introduce a family of photon-added as well as photon-depleted coherent states related to the inverse of ladder operators acting on hypergeometric coherent states. Their squeezing and antibunching properties are investigated in both conventional (nondeformed) and deformed quantum optics.

Analytic dynamics of the Morse oscillator derived by semiclassical closures.

Abstract: The quantized Hamilton dynamics methodology [O. V. Prezhdo and Y. V. Pereverzev, J. Chem. Phys. 113, 6557 (2000)] is applied to the dynamics of the Morse potential using the SU(2) ladder operators. A number of closed analytic approximations are derived in the Heisenberg representation by performing semiclassical closures and using both exact and approximate correspondence between the ladder and position-momentum variables. In particular, analytic solutions are given for the exact classical dynamics of the Morse potential as well as a second-order semiclassical approximation to the quantum dynamics. The analytic approximations are illustrated with the O–H stretch of water and a Xe–Xe dimer. The results are extended further to coupled Morse oscillators representing a linear triatomic molecule. The reported analytic expressions can be used to accelerate classical molecular dynamics simulations of systems containing Morse interactions and to capture quantum-mechanical effects.

The Supersymmetry Approaches to the Non-central Kratzer Plus Ring-Shaped Potential

Abstract: In this paper, we study the Schrodinger equation with non-central modified Kratzer potential plus a ring-shaped like potential, which is not spherically symmetric. We connect the corresponding Schrodinger equation to the Laguerre and Jacobi equations. These lead us to have some raising and lowering operators which are first order equations. We take advantage from these first order equations and discuss the supersymmetry algebra. And also we obtain the corresponding partner Hamiltonian for Kratzer potential and investigate the commutation relation for the generators algebra.

The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics

Abstract: Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.

Ideals and structure of operator algebras

Abstract: We continue the study of r-ideals, `-ideals, and HSA's in operator algebras. Some applications are made to the structure of operator algebras, including Wedderburn type theorems for a class of operator algebras. We also consider the one-sided M-ideal structure of certain tensor products of operator algebras.

Global estimates for singular integrals of the composite operator

Abstract: We establish the Poincare-type inequalities for the composition of the homotopy operator and the projection operator applied to the nonhomogeneous $A$-harmonic equation in John domains. We also obtain some estimates for the integral of the composite operator with a singular density.

Ladder operators and coherent states for continuous spectra

Abstract: The notion of ladder operators is introduced for systems with continuous spectra. We identify two different kinds of annihilation operators allowing the definition of coherent states as modified 'eigenvectors' of these operators. Axioms of Gazeau?Klauder are maintained throughout the construction.

Multiple operator integrals and spectral shift

Abstract: Multiple scalar integral representations for traces of operator derivatives are obtained and applied in the proof of existence of the higher order spectral shift functions.

Multiparameter deformation theory for quantum confined systems

Abstract: We introduce a generalized multiparameter deformation theory applicable to all supersymmetric and shape-invariant systems. Taking particular choices for the deformation factors used in the construction of the deformed ladder operators, we show that we can generalize the one-parameter quantum-deformed harmonic oscillator models and build alternative multiparameter deformed models that are also shape invariant like the primary undeformed system.

On the similarity of a J -nonnegative Sturm-Liouville operator to a self-adjoint operator

Abstract: In terms of Weyl-Titchmarsh m-functions, we obtain a new necessary condition for an indefinite Sturm-Liouville operator to be similar to a self-adjoint operator. This condition is used to construct examples of J-nonnegative Sturm-Liouville operators with singular critical point zero.