Showing papers on "Ladder operator published in 1992"

Operator Methods in Quantum Mechanics

Abstract: Preface The mathematical formalism of quantum mechanics The harmonic oscillator Other one-dimensional systems Angular momentum Spherically symmetric potentials Applications Factorization with application to the momentum representation The isotropic harmonic oscillator in an angular momentum basis The Coulomb problem in an angular momentum basis The Coulomb problem in the basis /HA OzL Tz Relativistic quantum mechanics Relativistic spherically symmetric problems References Appendices.

Fully symmetric operator spaces

Abstract: It is shown that certain interpolation theorems for non-commutative symmetric operator spaces can be deduced from their commutative versions. A principal tool is a refinement of the notion of Schmidt decomposition of a measurable operator affiliated with a given semi-finite von Neumann algebra.

The fokker-planck operator as an asymptotic limit

Abstract: It is shown that the Fokker-Planck operator describing a highly peaked scattering process in the linear transport equation is a formal asymptotic limit of the exact integral operator. It is also shown that such peaking is a necessary, but not sufficient, condition for the Fokker-Planck operator to be a legitimate description of such scattering. In particular, the widely used Henyey-Greenstein scattering kernel does not possess a Fokker-Planck limit.

Exactly solvable potentials and quantum algebras.

Abstract: A set of exactly solvable one-dimensional quantum-mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials. A general solution includes Shabat's infinite number soliton system and leads to raising and lowering operators satisfying a q-deformed harmonic-oscillator algebra. In the latter case the energy spectrum is purely exponential and physical states form a reducible representation of the quantum conformal algebra ${\mathrm{su}}_{\mathit{q}}$(1,1).

Quadratic algebras and dynamics in curved spaces. II. The Kepler problem

Abstract: The symmetry aspects of the Kepler problem in a space of constant negative curvature are considered. It is shown that the algebra of the hidden symmetry reduces to the quadratic Racah algebraQR(3), and this makes it possible to express the coefficients of the overlapping of the wave functions in the spherical and parabolic coordinates in terms of Wilson-Racah polynomials. It is shown that the dynamical symmetry algebra that generates the spectrum is the quadratic Jacobi algebraQJ(3). Its ladder operators permit explicit construction of wave functions in the coordinate representation with the ground state as the starting point.

Deformed conformal and supersymmetric quantum mechanics

Abstract: Within the standard quantum mechanics a q-deformation of the simplest N=2 supersymmetry algebra is suggested. Resulting physical systems do not have conserved charges and degeneracies in the spectra. Instead, superpartner Hamiltonians are q-isospectral, i.e., the spectrum of one can be obtained from another (with possible exception of the lowest level) by the q2-factor scaling. A special class of the self-similar potentials is shown to obey the dynamical conformal symmetry algebra suq(1,1). These potentials exhibit exponential spectra and corresponding raising and lowering operators satisfy the q-deformed harmonic oscillator algebra of Biedenharn and Macfarlane.

Certain classes of extension of a lacunary Hermitian operator

Abstract: Certain classes of extensions of a lacunary Hermitian operator are described in terms of abstract boundary conditions. The connection between the asymptotic behavior of eigenvalues of an extension near the boundary of lacuna and the asymptotic of the negative spectrum of the corresponding boundary operator is found.

Limiting Procedures for the Optical Phase Operator

Abstract: We examine some of the attempts to describe the phase of a single field mode by a quantum operator acting in the conventional infinite Hilbert space. These operators lead to bizarre properties such as non-random phases for the number states and experience consistency difficulties when used to obtain a phase probability density. Moreover, in these approaches operator functions of phase are not simply functions of a phase operator. We show that these peculiarities do not arise when the Hermitian optical phase operator is employed. In our opinion, the problems associated with the descriptions of phase in conventional infinite Hilbert space arise from the nature of the limiting process.

Phase operator formalism for a two-mode photon system

Abstract: The phase operator for a two-mode photon system is defined in terms of the relative-number states, and its property is investigated in detail. The phase operator thus defined has a unitary exponential form. It is shown that the average value and fluctuation of the phase operator, where one mode is in some physical state, such as a coherent or a squeezed state, and the other is in a vacuum state, are equivalent to those obtained by means of the Pegg–Barnett phase operator for a single-mode photon. The average value and fluctuation of the phase operator are calculated for coherent and squeezed states of a two-mode photon system.

The time-evolution operator for higher-order singular lagrangians

Abstract: A generalization of the time-evolution operator for higher-order singular-Lagrangian systems is developed and used for the study of the relations between Lagrangian and Hamiltonian formalisms, and, more specifically, for establishing the classification of constraints on both systems.

Relativistic internal time operator

Abstract: We construct a self-adjoint time operator for massless relativistic systems in terms of the generators of the Poincare group The Lie algebra generated by the time operator and the generators of the Poincare group turns out to be an infinitedimensional extension of the Poincare algebra The internal time operator generates two new entities, namely the velocity operator and the internal position operator The transformation properties of the internal time and position operator under Lorentz boosts are different from what one would expect from relativity theory This difference reflects the fact that the time concept associated with the internal time operator is radically different from the time coordinate of Minkowski space, due to the nonlocality of the time operator The spectral projections of the time operator allow us to construct incoming subspaces for the wave equation without invoking Huygens' principle, as in two and one spatial dimensions where Huygens' principle does not hold

The Weyl Quantization of Phase Angle

Abstract: We suggest a candidate for a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule. The matrix elements of this operator, for harmonic oscillator number states, coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.

Properties of the Dirac-Weyl operator with a strongly singular gauge potential

Abstract: Considered is a quantum system of a charged particle moving in the plane R2 under the influence of a perpendicular magnetic field concentrated on some fixed isolated points in R2. Such a magnetic field is represented as a finite linear combination of the two‐dimensional Dirac delta distributions and their derivatives, so that the gauge potential of the magnetic field also may be strongly singular at those isolated points. Properties of the Dirac–Weyl operator with such a singular gauge potential are investigated. It is seen that some of them depend on whether the magnetic flux is locally quantized or not. Particular attention is paid to the zero‐energy state. For each of the self‐adjoint realizations of the Dirac–Weyl operator, the number of the zero‐energy states is computed. It is shown that, in the present case, a theorem of Aharonov and Casher [Phys. Rev. A 19, 2461 (1979)], which relates the total magnetic flux to the number of zero‐energy states, does not hold. It is also proven that the spectrum of...

Operator Means and the Relative Operator Entropy

Abstract: The notion of operator monotone functions was introduced by Lowner and that of operator concave functions by Kraus who is his student. Operator means were introduced by Ando and the general theory of them was established by Kubo and Ando himself. By their theory, a nonnegative operator monotone function is now considered as a variation of an operator mean. However this theory does not include the logarithm and the entropy function which are operator monotone and often used in information theory. These functions are operator concave and satisfy Jensen’s inequality. So, considering operator means from the historical viewpoint, we shall introduce the relative operator entropy by generalizing the Kubo-Ando theory. Though its definition is derived from the Kubo-Ando theory of operator means, it can be constructed also in some ways. The relative operator entropy has of course some entropy-like properties.

Explicit canonical tensor operators and orthonormal coupling coefficients of SU(3)

Abstract: The canonical unit SU(3) tensor operators are constructed by means of the stretched coupling of the auxiliary maximal and minimal null space tensor operators, with the renormalization factors expressed in terms of the denominator functions of Biedenharn, Gustafson, Lohe, Louck, and Milne. The matrix elements of the maximal null space tensor operators are expressed with the help of the modified projection operators of Asherova and Smirnov. The self‐conjugate minimal null space tensor operators are expressed in terms of the group generators with the help of the weight lowering operator technique. The corresponding extreme isoscalar factors of the Clebsch–Gordan (Wigner) coefficients are used as constructive elements of the explicit recursive expression for the general orthonormal isoscalar factors of SU(3) with its considerable simplication for the boundary values of parameters. The general isofactors are also expanded in the different ways in terms of their boundary values. The new classes of the generaliz...

Algebraic approach to matrix elements: Recurrence relations and closed formulas for hydrogenlike wave functions.

Abstract: An algebraic approach to obtain ladder operators for hydrogenlike wave functions from the algebraic representation of the ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ Laguerre associated polynomials directly involved is presented. The method leads to diverse ladder operators depending on the manner as the operational form of ${\mathit{L}}_{\mathit{n}}^{\mathit{k}}$ is used. As an example, this work deals with linear creation and annihilation operators of two kinds: some that influence the n energy quantum number and others acting on the l orbital number according to usual ladder properties. In addition, as a useful application of the raising and lowering operators thus obtained, generalized recurrence relations and closed formulas for the calculation of multipole matrix elements are algebraically deduced by means of a procedure that takes advantage of the relationship between the ladder operators associated with the bra and those related to the ket.

Supersymmetric Quantum Theory

Abstract: We recall the treatment of the harmonic oscillator by means of ladder operators a and a † and pose the following question: Can one also represent other Hamiltonians as the “absolute square” of an operator and then construct their solutions algebraically?

Statistical Properties of a New Squeezed Operator Model

Abstract: We introduce a new squeezed operator which is a combination of the two-mode squeezed operator and the two single photon squeezed operator. Here we study the effects of this operator on the photon number sum and difference. The squeezing in the frequency converter model has been examined, and statistical investigations are carried out for the quasi-probability distribution functions (Wigner function and Q-function). An application to the parametric amplifier is given.

Alternative approach to the factorization method

Abstract: A procedure to obtain the operational solutions of second order differential equations related with Sturm–Liouville problems is presented. The method is based on the commutation relation between the ladder operators themselves, with a certain structure, and the position and momentum operators. Even though the creation and annihilation operators, derived by the proposed approach, factorize as expected the corresponding differential equation, the method does not use, as original premise factoring, the differential relation under consideration. That is, the displayed procedure is quite different, simple, and direct when compared with other procedures such as the factorization method of Infeld and Hull. To illustrate the above, the usefulness of the proposed procedure is shown by finding the ladder operators associated to the quantum numbers n and I for various potential wave functions. © 1992 John Wiley & Sons, Inc.

Sojourn-time operator approach to interaction time in quantum scattering: General formulation for arbitrary scattering systems

Abstract: A general formulation of the sojourn-time operator approach is presented, valid for arbitrary scattering systems and free of restrictions to one-dimensional systems present in the previous work of the authors. We show how first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. Both time-dependent and time-independent perturbations, including coupling between translation and internal degrees of freedom, are considered. The mean values of these Hermitian sojourn-time operators give the mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system defined by sets of commuting observables. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, to a first-order approximation, fully expressible in terms of these operators. We also show that this approach provides a unified treatment of the various idealized one-dimensional systems, which have frequently been employed in the debate on tunneling and interaction times. In particular, we provide some insight into the role of complex times, which appear as combinations of matrix elements of the sojourn-time operators.

Quantum mechanics in Riemannian spacetime. II. Operators of observables

Abstract: The formulation of the generally covariant analog of standard (nonrelativistic) quantum mechanics in a general Riemannian spacetime begun in earlier studies of the author is continued with the introduction of asymptotic (with respect toc−2) operators of the spatial position of a spinless particle and of the projection of its momentum onto an arbitrary spacetime direction. The connection between the position operator and the generalization of theV1,3 Newton—Wigner operator is established. It is shown that the projection of the momentum onto the 4-velocity of the frame of reference (the energy operator) is unitarily equivalent to the Hamiltonian in the Schrodinger equation.

QPFT operator algebras and commutative exterior differential calculus

Abstract: The reduction of the structure theory of the operator algebras of quantum projective (sl(2, ℂ)-invariant) field theory (QPFT operator algebras) to a commutative exterior differential calculus by means of the operation of renormalization of a pointwise product of operator fields is described

The time evolution operator of an atom in a deformed intense field

Abstract: We explicitly calculate the time evolution operator of an atom in a deformed electromagnetic field which is dependent on deformation parameter and possesses quantum group symmetry. The density operator and the time variation of photon distribution are also given, both with and without intensity-dependent coupling.

Scattering by periodically moving obstacles

Abstract: Suppose , is a homogeneous hyperbolic matrix, is the operator taking the Cauchy data for the system for into the corresponding data at time , and is the analogous operator constructed from the exterior mixed problem for the hyperbolic system . It is assumed that the boundary of the domain and the coefficients of the operator are periodic in with period , for , the noncapturing condition is satisfied, the matrix is elliptic, and the energy of solutions of the exterior problem is uniformly bounded for .Under these conditions it is proved that the space generated by the eigenfunctions of the monodromy operator with eigenvalues on the unit circle is finite dimensional; for initial data with compact support the asymptotics of the solution of the exterior problem as is obtained; in particular, it is shown that , , where is the operator of projection onto ; and existence of the wave operators constructed on the basis of and and of the scattering operator is proved.