Showing papers on "Ladder operator published in 1994"
Book•
01 Jan 1994
TL;DR: In this paper, the Lanczos tridiagonalization procedure was used to construct quasi-exactly solvable models with separable variables, and the Gelfand-Levitan equation was used for the first time.
Abstract: QUASI-EXACT SOLVABILITY-WHAT DOES THAT MEAN? Introduction Completely algebraizable spectral problems The quartic oscillator The sextic oscillator Non-perturbative effects in an explicit form and convergent perturbation theory Partial algebraization of the spectral problem The two-dimensional harmonic oscillator Completely integrable quantum systems Deformation of completely integrable models Quasi-exact solvability and the Gaudin model The classical multi-particle Coulomb problem Classical formulation of quantal problems The Infeld-Hull factorization method and quasi-exact solvability The Gelfand-Levitan equation Summary Historical comments SIMPLEST ANALYTIC METHODS FOR CONSTRUCTING QUASI-EXACTLY SOLVABLE MODELS The Lanczos tridiagonalization procedure The sextic oscillator with a centrifugal barrier The electrostatic analogue-the quartic oscillator Higher oscillators with centrifugal barriers The electrostatic analogue-the general case The inverse method of separation of variables The Schrodinger equations with separable variables Multi-dimensional models The "field-theoretical" case Other quasi-exactly solvable models THE INVERSE METHOD OF SEPARATION OF VARIABLES Multi-parameter spectral equations The method-general formulation The case of differential equations Algebraically solvable multi-parameter spectral equations An analytic method Reduction to exactly solvable models The one-dimensional case-classification Elementary exactly solvable models The multi-dimensional case-classification CLASSIFICATION OF QUASI-EXACTLY SOLVABLE MODELS WITH SEPARATE VARIABLE Preliminary comments The one-dimensional non-degenerate case The non-degenerate case-the first type The non-degenerate case-the second type The non-degenerate case-the third type The one-dimensional simplest degenerate case The simplest degenerate case-the first type The simplest degenerate case-the second type The simplest degenerate case-the third type The one-dimensional twice-degenerate case The twice-degenerate-the first type The twice-degenerate case-the second type The one-dimensional most degenerate case The multi-dimensional case COMPLETELY INTEGRABLE GAUDIN MODELS AND QUASI-EXACT SOLVABILITY Hidden symmetries Partial separation of variables Some properties of simple Lie algebras Special decomposition in simple Lie algebras The generalized Gaudin model and its solutions Quasi-exactly solvable equations Reduction to the Schrodinger form Conclusions Appendices A: The Inverse Schrodinger Problem and Its Solution for Several Given States Appendices B: The Generalized Quantum Tops and Exact Solvability Appendices C: The Method of Raising and Lowering Operators Appendices D: Lie Algebraic Hamiltonians and Quasi-Exact Solvability References Index
591 citations
TL;DR: In this article, a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model is presented, with the use of the boost operator, and a simple description of conserved charges is found in terms of a Catalan tree.
Abstract: We present a detailed analysis of the structure of the conservation laws in quantum integrable chains of the XYZ-type and in the Hubbard model. With the use of the boost operator, we establish the general form of the XYZ conserved charges in terms of simple polynomials in spin variables and derive recursion relations for the relative coefficients of these polynomials. For two submodels of the XYZ chain - namely the XXX and XY cases, all the charges can be calculated in closed form. For the XXX case, a simple description of conserved charges is found in terms of a Catalan tree. This construction is generalized for the su(M) invariant integrable chain. We also indicate that a quantum recursive (ladder) operator can be traced back to the presence of a hamiltonian mastersymmetry of degree one in the classical continuous version of the model. We show that in the quantum continuous limits of the XYZ model, the ladder property of the boost operator disappears. For the Hubbard model we demonstrate the non-existence of a ladder operator. Nevertheless, the general structure of the conserved charges is indicated, and the expression for the terms linear in the model's free parameter for all charges is derived in closed form.
83 citations
18 citations
TL;DR: In this paper, the orthogonality-constrained one-body eigenvalue problem is formulated for complete, finite-dimensional spaces in terms of projection operators that partition the space into two orthogonal subspaces, one containing the constraints and the other containing the desired solutions.
Abstract: The orthogonality-constrained one-body eigenvalue problem is formulated for complete, finite-dimensional spaces in terms of projection operators that partition the space into two orthogonal subspaces, one containing the constraints and one containing the desired solutions. We derive a Hermitian operator, associated with the Hamiltonian, having eigenvectors that are correspondingly partitioned, each subset of the eigenvectors providing a basis for one of the subspaces. This Hermitian operator advantageously replaces a non-Hermitian operator that was proposed in recent work of this group. The non-zero spectrum of the two operators is demonstrably the same, and the occurrence of null eigenvalues is clarified through characterization of the associated eigenvectors. The deviation from zero of these eigenvalues (apart from numerical inaccuracies) is related directly to the normalization of the constraint vectors. The formalism is extended to the case of non-orthonormal primitive basis sets, and a numerical application is carried out using B-spline functions.
16 citations
TL;DR: In this article, the wave functions of fermions and a Dirac operator on quantum-two spheres can be constructed in a manifestly covariant way under the quantum group SU(2)q.
Abstract: We investigate the spin-1/2 fermions on quantum-two spheres. It is shown that the wave functions of fermions and a Dirac operator on quantum-two spheres can be constructed in a manifestly covariant way under the quantum group SU(2)q. The concept of total angular momentum and chirality can be expressed by using q-analog of Pauli-matrices and appropriate commutation relations.
16 citations
TL;DR: In this paper, the authors defined coherent states in deformed quantum mechanics as eigenstates of the annihilation operator of q-algebras and associated them with elements in a Hilbert space of analytic functions.
12 citations
TL;DR: A new, operator‐based algorithm for the calculation of Clebsch–Gordon coefficients for the group SU 3 is introduced, and an A N S I C program for its implementation is described.
Abstract: A new, operator‐based algorithm for the calculation of Clebsch–Gordon coefficients for the group SU 3 is introduced, and an A N S I C program for its implementation is described. The algorithm involves the use of raising and lowering operators in analogy to a standard pedagogical approach to the more familiar SU 2 Clebsch–Gordon coefficients. The nature of the approach makes the sign conventions utilized and the rules for resolution of the outer degeneracy explicit, and therefore allows straightforward changes to adapt the program to other choices. The program is written to allow its execution on personal computers of modest memory size, but also to take advantage of available memory allowing more complex results to be obtained on larger computing systems.
9 citations
TL;DR: In this article, the structure of the finite-dimensional representations of the quantum superalgebra U q (osp(1/2)) is considered and the projection operator for this quantum super algebra is derived.
Abstract: The structure of the finite‐dimensional representations of the quantum superalgebra U q (osp(1‖2)) is considered and the projection operator for this quantum superalgebra is derived. Application of the projection operator permits one to obtain an explicit analytical formula for the q analog of the Clebsch–Gordan coefficients. Pseudo‐orthogonality relations and some other properties of Clebsch–Gordan coefficients are given.
9 citations
TL;DR: This work investigates the most general form of the three-body mass operator in the instant form of relativistic dynamics and discusses the dependence of this operator on the total momenta of each pair of quarks in the c.m. frame of three quarks.
Abstract: We investigate the most general form of the three-body mass operator in the instant form of relativistic dynamics. It is shown that this operator is defined not only by the two-body mass operators and the three-body interaction operator but also by some three extra unitary operators. The latter are nontrivial already on the one-gluon-exchange level and, therefore, must be present in any relativistic constituent quark model. The restrictions imposed by the relativistic invariance on the form of the two-body energy operator in first order in 1/[ital m][sup 2] are discussed in detail. We write down the explicit expression for the three-quark mass operator in this order and discuss the dependence of this operator on the total momenta of each pair of quarks in the c.m. frame of three quarks.
7 citations
TL;DR: In this paper, it was shown that the construction of the class operator for SU(2) is a partial case of a much more general problem, that of decomposing an operator into components transforming under conjugation according to a given irreducible representation.
Abstract: It is shown that the construction of the class operator for SU(2) is a partial case of a much more general problem, that of decomposing an operator into components transforming under conjugation according to a given irreducible representation. The problem is solved generally for arbitrary compact groups and some possibilities for extensions of this procedure to the case of non-compact groups are indicated.
7 citations
TL;DR: In this paper, the authors examined the possibility of obtaining the transference of the squeezing effect between two coupled oscillators, one of them described by a quadratic Hamiltonian in terms of the ladder operators, the other one being a linear harmonic oscillator, plus an interaction term.
Abstract: We examine the possibility of obtaining the transference of the squeezing effect between two coupled oscillators, one of them described by a quadratic Hamiltonian in terms of the ladder operators, the other one being a linear harmonic oscillator, plus an interaction term. We obtain an exact solution for the time evolution of our coupled system which allows us to find the variances for one and two-mode oscillations. It is shown that the squeezing generated in one of the oscillators may or may not spread to the other oscillator, depending on the choice of the involved parameters. Other interesting features exhibited for the one- and two-mode oscillations are also discussed.
TL;DR: In this article, the role of similarity transformations is emphasized and the connection between the Lie algebra and similarity transformation is made explicit, leading to a particularly simple solution whenever the Hamiltonian can be expressed as a linear function of the generators.
Abstract: Bound‐state solutions of several different Schrodinger equations are calculated rather efficiently using the methods of Lie algebra. In all cases considered, either of the algebras SO(3) or SO(2,1) provides a suitable framework, and there is no advantage to either choice. However, the choice of an appropriate realization of the generators is of greater significance, leading to a particularly simple solution whenever the Hamiltonian can be expressed as a linear function of the generators. In certain cases, the Hamiltonian can be expressed only as a bilinear function of the generators and only part of the bound‐state spectrum can be calculated analytically to yield a finite set of so‐called quasiexact solutions. These may be interrelated by means of suitable ladder operators and it is possible, though by no means necessary, to adopt a particular finite‐dimensional representation of the underlying algebra. The present work emphasizes the role of similarity transformations, and the connection between the Lie ...
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TL;DR: In this article, it was shown that despite the non-local dependence of the energy density on the Klein-Gordon operator, the commutators of the physical observables vanish for space-like separations.
Abstract: The square-root Klein-Gordon operator,√(m^2− ∇^2), is a non-local operator with a natural scale inversely proportional to the mass (the Compton wavelength). There is no fundamental reason to exclude negative energy states from a “square-root” propagation law. We find several possible Hamiltonians associated with √(m^2− ∇^2) which include both positive and negative energy plane wave states. It is possible to satisfy the equations of motion with commutators or anticommutators. For the scalar case, only the canonical commutation rules yield a stable vacuum. We investigate microscopic causality for the commutator of the Hamiltonian density. We find that despite the non-local dependence of the energy density on the field operators, the commutators of the physical observables vanish for space-like separations. Hence, Pauli’s result can be extended to the non-local case. Pauli explicitly excluded √(m^2− ∇^2) because this operator acts non-locally in the coordinate space. The Mandelstam representation offers the possibility of avoiding the difficulties inherent in minimal coupling (Lorentz invariance and gauge invariance). We also compute the propagators for the scattering problem and investigate thesolutions of the square-root equation in the Aharonov-Bohm problem.
TL;DR: In this article, it was shown that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and every non-algebraic normal operator with a two-dimensional extension which fails to have a cycle-commutant has a separating vector.
Abstract: We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.
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TL;DR: In this article, the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to biorthogonal rational functions were studied.
Abstract: We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szegő and for their four parameter generalization to ${}_4\phi_3$ biorthogonal rational functions on the unit circle.
TL;DR: In this paper, the exact solutions of the photon annihilation operator a( t), the atomic dipole moment operator σ-( t), and the population difference operator ρz ( t) are obtained.
Abstract: The dynamics of the degenerate two-photon Jaynes-Cummings model in Heisenberg picture is investigated By solving the Heisenberg equations, the exact solutions of the photon annihilation operator a( t), the atomic dipole moment operator σ-( t), and the population difference operator σz ( t) are obtained We demonstrate that the correlation functions for the photon annihilation and dipole moment operators can be directly evaluated from these exact solutions The solutions of the time-evolution operators are shown to be quite useful in the calculation of two-time correlation functions Furthermore, some results of the dynamical behavior of the average values for the photon number , the in-phase field component , and the population difference are demonstrated by computer simulations
TL;DR: In this paper, the quantum deformation algebra SOq(2,1) is applied to derive the q-analogues of the ladder and shift operators for the radial Coulomb, radial harmonic oscillator and Morse oscillator potentials.
Abstract: The quantum deformation algebra SOq(2,1) is studied and applied to derive the q-analogues of the ladder and shift operators for the radial Coulomb, radial harmonic oscillator and Morse oscillator potentials. The q-deformed operators in all three cases are found to act like shift operators, called q-shift operators. Their possible similarity with the quasi-shift operators arising in supersymmetric quantum mechanics, or factorization, of the radial harmonic oscillator is also pointed out.
TL;DR: In this paper, a phase operator formalism is presented within the framework of thermo field dynamics (TFD), where a unitary phase operator (or phase factor) can be defined as a canonical conjugate of the generator of phase shift in TFD since the generator has a lower bounded spectrum.
Abstract: A phase operator formalism is presented within the framework of thermo field dynamics (TFD). It is shown that a unitary phase operator (or phase factor) can be defined as a canonical conjugate of the generator of phase shift in TFD since the generator has a lower bounded spectrum. The tilde conjugation plays an essential role in defining the unitary phase operator in TFD. The unitary phase operator is expressed in terms of the relative-number states. The properties of the unitary phase operator are investigated in detail. The relations to the Pegg-Barnett phase operator and to the polar decomposition of the operator which appears in the heterodyne detection are found. Furthermore, by making use of the phase operator method, a quantum phase measurement is considered and the relation to the optimal probability operator measure in the conventional theory is investigated.
TL;DR: In this paper, the problem of constructing the electromagnetic-current operator for a relativistic system with a fixed number of constituents was considered and the explicit solution of the problem to order 1/c4 was given.
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TL;DR: In this article, a generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems, which leads to compatibility of quantum equations of motion and canonical commutation relations.
Abstract: The dissipative models in string theory can have more broad range of application: 1) Noncritical strings are dissipative systems in the "coupling constant" phase space. 2) Bosonic string in the affine-metric curved space is dissipative system.
But the quantum descriptions of the dissipative systems have well known ambiguities. In order to solve the problems of the quantum description of dissipative systems we suggest to introduce an operator W in addition to usual (associative) operators. The suggested operator algebra does not violate Heisenberg algebra because we extend the canonical commutation relations by introducing an operator W of the nonholonomic quantities in addition to the usual (associative) operators of usual (holonomic) coordinate -momentum functions. To satisfy the generalized commutation relations the operator W must be nonassociative nonLieble (does not satisfied the Jacobi identity) operator. As the result of these properties the total time derivative of the multiplication and commutator of the operators does not satisfies the Leibnitz rule. This lead to compatibility of quantum equations of motion for dissipative systems and canonical commutation relations. The suggested generalization of the von Neumann equation is connected with classical Liouville equation for dissipative systems.
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TL;DR: In this paper, a closed expression for the density operator of the damped harmonic oscillator is extracted from the master equation based on the Lindblad theory for open quantum systems and the entropy and effective temperature of the system are subsequently calculated and their temporal behaviour is surveyed by showing how these quantities relax to their equilibrium values.
Abstract: A closed expression for the density operator of the damped harmonic oscillator is extracted from the master equation based on the Lindblad theory for open quantum systems The entropy and effective temperature of the system are subsequently calculated and their temporal behaviour is surveyed by showing how these quantities relax to their equilibrium values The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function
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TL;DR: In this paper, the shift operator is constructed from the cyclic quantum dilogarithm with q is a root of unit and the representation is given for the current algebra introduced by Faddeev et al.
Abstract: From the cyclic quantum dilogarithm the shift operator is constructed with q is a root of unit and the representation is given for the current algebra introduced by Faddeev et al. It is shown that the theta-function is factorizable also in this case by using the star-square equation of the Baxter-Bazhanov model.
TL;DR: In this article, the authors present three operators in quantum mechanics that obey the commutation relations of quantum group SUq(2) and are called the quantumq-analog angular momentum operators.
Abstract: We present three operators in quantum mechanics that obey the commutation relations of quantum groupSUq(2). These operators are nonlinear combinations of the conventional angular momentum operators and are called the quantumq-analog angular momentum operators. When the quantum deformation parameterr = Inq vanishes, these quantumq-analog angular momentum operators reduce to the usual angular momentum operators.
01 May 1994
TL;DR: In this paper, the authors propose a method for obtaining generalized minimum-uncertainty squeezed states, giving examples, and relates it to known concepts, such as ladder-operator and displacement-operator squeezed states.
Abstract: Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the (1) displacement operator, (2) annihilation- (or ladder-) operator, and (3) minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known concepts, the author proposes a method for obtaining generalized minimum-uncertainty squeezed states, gives examples, and relates it to known concepts. He comments on the remaining concept, that of general displacement-operator squeezed states.
TL;DR: In this paper, a self-adojoint Usawa operator for the channel flow problem is studied. But this operator is used in the case of finite element discretisations and it is not considered in this paper.
Abstract: In this article we study the operator which appears in the Usawa algorithm classically used in the solution of the Stokes problem, in particular when finite element discretisations are used (see [1],[11]). We call this operator the Usawa operator. We consider S (and-Δ) as a self–adojoint operator in L2Ωfor several domains Ωwith a particular emphasis on the channel flow problem; in this case we determine all the eigenvalues of S and give a precise description of S.