Showing papers on "Ladder operator published in 2002"
TL;DR: In this paper, a realization of the raising and lowering operators for the Morse potential is presented, and it is shown that these operators satisfy the commutation relations for the SU(2) group.
Abstract: A realization of the raising and lowering operators for the Morse potential is presented. It is shown that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are obtained for the matrix elements of different operators such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied and an approach previously proposed to calculate the Franck–Condon factors is discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
167 citations
TL;DR: A closed form of the normalization constants of the wave function for the modified Poschl-Teller (MPT) potential is obtained from two different methods in this article, where it is shown that the discrete spectrum of the MPT potential is associated to the su(2) algebra.
Abstract: A closed form of the normalization constants of the wave function for the modified Poschl–Teller (MPT) potential is obtained from two different methods. It is shown that the discrete spectrum of the MPT potential is associated to the su(2) algebra. This identification is achieved by means of a realization of raising and lowering operators in terms of the physical variable u=tanh (αx). Analytical expressions for the matrix elements derived from these operators are obtained for the functions and . The harmonic limit of the su(2) operators is also analyzed. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem 86: 265–272, 2002
88 citations
TL;DR: In this article, the authors propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity.
Abstract: We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six-vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .
73 citations
TL;DR: In this paper, the integral Volterra operator on the space L 2 (0, 1) is considered and the set of extended eigenvalues of a complex number is chosen to be integral operators.
Abstract: In this paper we consider the integral Volterra operator on the space L 2 (0; 1). We say that a complex number is an extended eigenvalue ofV if there exists a nonzero operator X satisfying the equation XV = V X . We show that the set of extended eigenvalues of V is precisely the interval (0;1) and the corresponding eigenvectors may be chosen to be integral operators as well.
54 citations
TL;DR: In this article, a realization of the ladder operators for the solutions to the Schrodinger equation with a pseudoharmonic oscillator in 2D is presented, and closed analytical expressions are evaluated for the matrix elements of some operators r(2) and rpartial derivative/partial derivativer.
Abstract: A realization of the ladder operators for the solutions to the Schrodinger equation with a pseudoharmonic oscillator in 2D is presented. It is shown that those operators satisfy the commutation relations of an SU(1, 1) algebra. Closed analytical expressions are evaluated for the matrix elements of some operators r(2) and rpartial derivative/partial derivativer.
52 citations
TL;DR: In this paper, the Schrodinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric, gravitational field and electromagnetic field is analyzed.
Abstract: We analyse the Schrodinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric, gravitational field and electromagnetic field. We approach the Schrodinger operator in three independent ways: in terms of covariant differentials induced by the quantum connection, via a quantum Lagrangian and directly by the only requirement of general covariance. In particular, in the flat case, our Schrodinger operator coincides with the standard one.
43 citations
TL;DR: In this article, a realization of the raising and lowering operators for the Morse potential is presented, and it is shown that these operators satisfy the commutation relations for the SU(2) group.
Abstract: A realization of the raising and lowering operators for the Morse potential is presented. We show that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expre...
42 citations
TL;DR: In this article, it was shown that the Nth power of the lightcone evolution operator of the 2N-periodic quantum discrete Liouville model can be identified with the Dehn twist operator in quantum Teichmuller theory.
Abstract: It is shown that the Nth power of the light-cone evolution operator of the 2N-periodic quantum discrete Liouville model can be identified with the Dehn twist operator in quantum Teichmuller theory
40 citations
TL;DR: In this article, the Schrodinger equations for the Coulomb and the harmonic oscillator potentials are solved in the cosmic string conical spacetime, and the spherical harmonics with angular deficit are introduced.
Abstract: The Schrodinger equations for the Coulomb and the harmonic oscillator potentials are solved in the cosmic string conical spacetime. The spherical harmonics with angular deficit are introduced. The algebraic construction of the harmonic oscillator eigenfunction is performed through the introduction of non-local ladder operators. By exploring the hidden symmetry of the two-dimensional harmonic oscillator the eigenvalues for the angular momentum operators in three dimensions are reproduced. A generalization for N dimensions is performed for both Coulomb and harmonic oscillator problems in angular deficit spacetimes. The connection among the states and energies of both problems in these topologically non-trivial spacetimes is thus established.
37 citations
TL;DR: In this paper, a general theory of Schrodinger equations which are exactly solvable in terms of associated special functions is presented. But it is not a theory of hypergeometric type equations.
Abstract: The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in a supersymmetric approach to these Hamiltonians are consequences of some formulae concerning the general theory of associated special functions. We use this connection in order to obtain a general theory of Schrodinger equations exactly solvable in terms of associated special functions, and to extend certain results known in the case of some particular potentials.
33 citations
TL;DR: In this article, the authors used dynamical invariant operator method to find the quantum mechanical solution of a harmonic plus inverse harmonic oscillator with time-dependent coefficients and constructed lowering and raising operators from the invariant operators.
Abstract: We used dynamical invariant operator method to find the quantum mechanical solution of a harmonic plus inverse harmonic oscillator with time-dependent coefficients. The eigenvalue of invariant operator is obtained and is constant with time. We constructed lowering and raising operators from the invariant operator. The solution of Schrodinger equation is obtained using operator method. We have also used ladder operators to obtain various expectation values of the time-dependent system. The results in this manuscript are not only more general than the existing results in the literatures but also well match with others.
TL;DR: In this article, a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions is constructed and the corresponding coherent states are investigated by adopting the Gazeau-Klauder approach, where resolution of unity and overlapping properties are examined.
Abstract: We explicitly construct a Hamiltonian whose exact eigenfunctions are the generalized Laguerre functions. Moreover, we present the related raising and lowering operators. We investigate the corresponding coherent states by adopting the Gazeau–Klauder approach, where resolution of unity and overlapping properties are examined. Coherent states are found to be similar to those found for a particle trapped in a Poschl–Teller potential of the trigonometric type. Some comparisons with Barut–Girardello and Klauder–Perelomov methods are noted.
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TL;DR: In this paper, the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge is defined, and the extension problems for vertex operator algebras can be described in terms of the associated modular band tensor categories.
Abstract: The notion of the genus of a quadratic form is generalized to vertex operator
algebras. We define it as the modular braided tensor category associated to a
suitable vertex operator algebra together with the central charge. Statements
similar as known for quadratic forms are formulated. We further explain how extension problems for vertex operator algebras can
bedescribed in terms of the associated modular braided tensor category.
TL;DR: In this article, a dissipative Schrodinger operator is regarded as a pseudo-Hamiltonian of an open one-dimensional quantum system and the carrier density is derived from the generalized eigenstate of the dissipative operator.
Abstract: We regard a current flow through an open one-dimensional quantum system which is determined by a dissipative Schrodinger operator. The imaginary part of the corresponding form originates from Robin boundary conditions with certain complex valued coefficients imposed on Schrodinger’s equation. This dissipative Schrodinger operator can be regarded as a pseudo-Hamiltonian of the corresponding open quantum system. The dilation of the dissipative operator provides a (self-adjoint) quasi-Hamiltonian of the system, more precisely, the Hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system. The carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. Finally a rigorous setup of a dissipative Schrodinger–Poisson system is outlined.
TL;DR: In this article, the central principle of affine quantum gravity is secured and maintaining the strict positivity of the matrix {ab(x) composed of the spatial components of the local metric operator, and a regularized version of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well defined functional integral involving only the same set of variables that appears in the usual classical formulation.
Abstract: The central principle of affine quantum gravity is securing and maintaining the strict positivity of the matrix {ĝab(x)} composed of the spatial components of the local metric operator. On spectral grounds, canonical commutation relations are incompatible with this principle, and they must be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the recently developed projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. It is anticipated that skills and insight to study this formulation can be developed by studying special, reduced-variable models that still retain some basic characteristics of gravity, specifically a partial second-class constraint operator structure. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models. Finally, developing a procedure to pass to the genuine physical Hilbert space involves several interconnected steps that require careful coordination.
TL;DR: In this article, it was shown that on standard operator algebras over spaces with dimension at least 2, the bijective solutions of that second equation are automatically additive.
Abstract: Jordan isomorphisms of rings are defined by two equations. The first one is the equation of additivity while the second one concerns multiplicativity with respect to the so-called Jordan product. In this paper we present results showing that on standard operator algebras over spaces with dimension at least 2, the bijective solutions of that second equation are automatically additive.
TL;DR: In this paper, a new general construction of recursion operator from zero curvature representation is presented, which is used for the stationary Nizhnik-Veselov-Novikov equation and a few low order symmetries generated with the help of this operator.
Abstract: We present a new general construction of recursion operator from zero curvature representation. Using it, we find a recursion operator for the stationary Nizhnik--Veselov--Novikov equation and present a few low order symmetries generated with the help of this operator.
TL;DR: In this paper, the authors obtained the differential operator realization for the annihilation operator A of generalized Heisenberg algebra corresponding to the given polynomial system, for which the matrix of the operator A in l 2 (Z + ) has only off-diagonal elements on the first upper diagonal different from zero.
Abstract: We obtain the differential operator realization for the annihilation operator A of generalized Heisenberg algebra corresponding to the given polynomial system. The important special case of orthogonal polynomial systems, for which the matrix of the operator A in l 2 ( Z + ) has only off-diagonal elements on the first upper diagonal different from zero, is considered. The known generalized Hermite polynomials give us an example of such an orthonormal system. The suggested approach can be applied to a similar investigation of various "deformed" polynomial systems.
TL;DR: In this article, it was shown that all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product, by using the inverse problem for quantum systems.
Abstract: Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.
TL;DR: For the quantum two-dimensional isotropic harmonic oscillator, it was shown in this article that the Infeld-Hull radial operators, as well as those of the supersymmetric approach for the radial equation, are contained in the constants of motion of the problem.
Abstract: For the quantum two-dimensional isotropic harmonic oscillator we show that the Infeld–Hull radial operators, as well as those of the supersymmetric approach for the radial equation, are contained in the constants of motion of the problem.
TL;DR: In this article, the properties of the modified Poschl-Teller potential are outlined and the harmonic limit for this system is discussed. And the expansion of the coordinate x and the momentum p from the operators of the SU(2) algebra are also obtained.
Abstract: The properties of the modified Poschl–Teller potential are outlined. The ladder operators are constructed directly from the wave functions without introducing any auxiliary variable. It is shown that these operators are associated to the SU(2) algebra. Analytical expressions for the functions sinh(αx) and (1/α)cosh(αx)d/dx are evaluated from these ladder operators. The harmonic limit for this system is discussed. The expansion of the coordinate x and the momentum p from the operators of the SU(2) are also obtained.
TL;DR: In this article, the relation between the operator function T and a selfadjoint self-adjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of T −T −1 is studied.
Abstract: We consider an operator function T in a Krein space which can formally be written as in (0.1) but the last term on the right of (0.1) is replaced by a relatively form-compact perturbation of a similar form. We study relations between the operator function T , a selfadjoint operator M in some Krein space, associated with T , and an operator which can be constructed with the help of the operator function −T −1. The results are applied to a Sturm-Liouville problem with a coefficient depending rationally on the eigenvalue parameter.
TL;DR: By taking advantage of dynamical invariant operator, this paper derived Schrodinger solution for exponentially damped harmonic oscillator with and without inverse quadratic potential, and investigated quantum mechanical energy expectation value, uncertainty relation, partition function and density operator of the system.
Abstract: By taking advantage of dynamical invariant operator, we derived Schrodinger solution for exponentially damped harmonic oscillator with and without inverse quadratic potential. We investigated quantum mechanical energy expectation value, uncertainty relation, partition function and density operator of the system. The various expectation values in thermal state are calculated using the diagonal element of density operator.
TL;DR: The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic as mentioned in this paper, and a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements is examined.
TL;DR: In this article, the eigenvalues and eigenfunctions of the Schrodinger equation with a nonrelativistic electron in a uniform magnetic field are presented, and a realization of the creation and annihilation operators for the radial wave-functions is carried out.
Abstract: The eigenvalues and eigenfunctions of the Schrodinger equation with a non-relativistic electron in a uniform magnetic field are presented. A realization of the creation and annihilation operators for the radial wave-functions is carried out. It is shown that these operators satisfy the commutation relations of an SU(1,1) group. Closed analytical expressions are evaluated for the matrix elements of different functions ρ2 and .
TL;DR: In this paper, the Hamilton operator is quadratic in these lowering and raising operators and corresponding representations of operators for action and angle are found, and the normally ordered representation of general operators using combinatorial elements such as partitions is derived.
Abstract: We define general lowering and raising operators of the eigenstates for one-dimensional quantum mechanical potential problems leading to discrete energy spectra and investigate their associative algebra. The Hamilton operator is quadratic in these lowering and raising operators and corresponding representations of operators for action and angle are found. The normally ordered representation of general operators using combinatorial elements such as partitions is derived. The introduction of generalized coherent states is discussed. Linear laws for the spacing of the energy eigenvalues lead to the Heisenberg–Weyl group and general quadratic laws of level spacing to unitary irreducible representations of the Lie group SU(1, 1) that is considered in detail together with a limiting transition from this group to the Heisenberg–Weyl group. The relation of the approach to quantum deformations is discussed. In two appendices, the classical and quantum mechanical treatment of the squared tangent potential is presented as a special case of a system with quadratic level spacing.
TL;DR: In this article, a wide class of q-deformed harmonic oscillators, including those of the Macfarlane type and Dubna type, can be described in a unified way.
Abstract: It is shown that a wide class of q-deformed harmonic oscillators, including those of the Macfarlane type and Dubna type, can be described in a unified way. The Hamiltonian of the oscillator is assumed to be given by a q-deformed anti-commutator of the q-deformed ladder operators. By solving q-difference equations, explicit coordinate representations of ladder operators and wave functions are derived, and unified parametric representations are found for q-Hermite functions and related formulas for oscillators of the Macfarlane and Dubna types. In addition to the well-known solutions with globally periodic structure, it is found that there exist an infinite number of solutions with globally aperiodic structure.
TL;DR: In this article, it was shown that the Landau quantum systems in a plane, sphere or hyperboloid can be explained in a complete and meaningful way by group-theoretical considerations concerning the symmetry group of the corresponding configuration space.
Abstract: We show that the Landau quantum systems (or integer quantum Hall effect systems) in a plane, sphere or a hyperboloid, can be explained in a complete and meaningful way by group-theoretical considerations concerning the symmetry group of the corresponding configuration space. The crucial point in our development is the role played by locality and its appropriate mathematical framework, the fibre bundles. In this way the Landau levels can be understood as the local equivalence classes of the symmetry group. We develop a unified treatment that supplies the correct geometric way to recover the planar case as a limit of the spherical or the hyperbolic quantum systems when the curvature goes to zero. This is an interesting case where a contraction procedure gives rise to nontrivial cohomology starting from a trivial one. We show how to reduce the quantum hyperbolic Landau problem to a Morse system using horocyclic coordinates. An algebraic analysis of the eigenvalue equation allows us to build ladder operators which can help in solving the spectrum under different boundary conditions.
TL;DR: In this paper, the Hopf term was constructed on fuzzy spheres and the corresponding basis set can be classified into different topological sectors by the magnetic quantum numbers, which can be used to construct BPS solitons.
Abstract: We formulate the $O(3) \s-$ model on fuzzy sphere and construct the Hopf term. We show that the field can be expanded in terms of the ladder operators of Holstein-Primakoff realisation of SU(2) algebra and the corresponding basis set can be classified into different topological sectors by the magnetic quantum numbers. We obtain topological charge $Q$ and show that $-2j\le Q \le2j$. We also construct BPS solitons. Using the covariantly conserved current, we construct the Hopf term and show that its value is $Q^2$ as in the commutative case. We also point out the interesting relation of physical space to deformed SU(2) algebra.