Showing papers on "Ladder operator published in 1991"
Book•
01 Jan 1991TL;DR: In this article, the authors formulate the absence theorem of phase transitions in its most general form within the C* setting and present a new axiomatic treatment of the construction of time evolutions and KMS states.
Abstract: This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made a considerable contribution. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of the C*-theory. The presentation, which is based on lectures given in Newcastle upon Tyne and Copenhagen, concentrates on topics involving quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, he globally constructs, within that setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment of the construction of time evolutions and KMS states.
182 citations
01 Mar 1991
TL;DR: In this paper, the authors give explicit internal definitions of the free product and the maximal operator-algebra tensor product of operator algebras and of the group operator algebra OA(G) of a discrete group.
Abstract: Using the characterization of unital operator algebras developed in [6], we give explicit internal definitions of the free product and the maximal operator-algebra tensor product of operator algebras and of the group operator algebra OA(G) of a discrete semigroup G (if G is a discrete group, then OA(G) coincides with the group C*-algebra C*(G)). This approach leads to new factorization theorems for polynomials in one and two variables.
71 citations
TL;DR: In this paper, a differential realization of so(2,1) is shown to be the potential algebra for the one-dimensional systems with the Morse or Gendenshtein potentials.
Abstract: A differential realization of so(2,1) is shown to be the potential algebra for the one-dimensional systems with the Morse or Gendenshtein potentials. This shows that two classes of Gendenshtein potentials will support the same eigenvalues as the Morse potential, and that the three sets of eigenfunctions may be derived in a common formalism. The potential algebra is then extended to a dynamical potential algebra with operators connecting states both in different potentials and with different energies, giving new dynamical algebras for the Gendenshtein problem. The matrix elements of certain corresponding operators in the three types of system may then be given by a single formula involving so(2,1) Wigner coefficients. The authors also give ladder operators connecting the Gendenshtein potential eigenstates.
59 citations
51 citations
TL;DR: In this paper, a relative number state (RNS) representation of a system composed of two distinguishable subsystems is proposed, and a phase variable as a quantum mechanical operator conjugate to the relative number operator is defined based on the RNS representation space.
Abstract: A relative number state (RNS) representation of a system composed of two distinguishable subsystems is proposed. A phase variable as a quantum mechanical operator conjugate to a relative number operator is defined based on the RNS representation space. The phase operator is expressed as a unitary exponential operator. The properties of the relative number operator, the phase operator, and their eigenstates are investigated in detail. The phase variable has a maximum uncertainty in any stationary state. Also, a time operator as a dynamical variable can be defined in the RNS representation space. The RNS representation is closely related to the Liouville space formulation and to thermofield dynamics. The RNS representation is shown to be a suitable method for investigating the Josephson junction with ultrasmall capacitance. A basic formulation of number‐phase quantization in the Josephson junction is given in terms of the RNS representation.
46 citations
TL;DR: In this paper, the Gelfand-Zetlin basis is adapted to SU(N)q forq a root of unit, where extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center.
Abstract: The Gelfand-Zetlin basis is adapted toSU(N)q forq a root of unit. Extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center. A crucial identity is derived and proved, which guarantees the periodicity of the action of the generators. Full periodicity is relaxed by stages, some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation. In the extreme case of highest weight representations. all the raising and lowering operators are nilpotent. As an alternative approach an auxiliary algebra giving all the periodic representations is presented. An explicit solution of this system forN=3, while fully equivalent to the G.-Z. basis, turns out to be much simpler.
41 citations
TL;DR: In this paper, the dynamics of the q-deformed quantum harmonic oscillator initially prepared in the qanalogue of the coherent state is analyzed. And the squeezing of the second-order moments of this operator is studied.
Abstract: We analyse the dynamics of the q-deformed quantum harmonic oscillator initially prepared in the q-analogue of the coherent state. Non-trivial behaviour of the mean values of the q-position operator is observed. The squeezing of the second-order moments of this operator is studied.
40 citations
TL;DR: In this article, the hypervirial theorem is used to derive closed-form expressions for diagonal and off-diagonal matrix elements for any values of the n and l quantum numbers and can be easily programmed on a hand calculator.
Abstract: Recurrence formulae determining radial matrix elements of rs, rs(1n r)k and rs dn/drn between hydrogenic wavefunctions are derived using ladder operators together with the hypervirial theorem. The closed-form expressions are well suited to computing diagonal and off-diagonal matrix elements for any values of the n and l quantum numbers and can easily be programmed on a hand calculator.
25 citations
TL;DR: In this paper, a phase variable as a quantum mechanical operator, which is canonically conjugate to the number-difference operator, is defined within the framework of the Liouville space formulation.
22 citations
17 citations
TL;DR: A representation of the oscillators for infinite statistics in terms of bosons is given and the relation to the Susskind-Glogower operator is discussed.
Abstract: A representation of the oscillators for infinite statistics in terms of bosons is given. The relation to the Susskind-Glogower operator is also discussed.
TL;DR: In this article, an explicit unitary scheme with arbitrary high order of accuracy was proposed to solve operator equations of motion in quantum mechanics and in quantum scalar field theory, and it was shown that the scheme can be used to solve the problem in any quantum system.
Abstract: To solve operator equations of motion in quantum mechanics and in a quantum scalar field theory, we propose an explicit unitary scheme with arbitrary high order of accuracy.
TL;DR: In this paper, the vibration-translation energy transfer in an atom-Morse oscillator collision was studied, and the vibration transition probability was evaluated as a function of the total collision energy, and also as the strength of the Morse potential.
Abstract: We study the vibration-translation energy transfer in an atom-Morse oscillator collision. We obtain the interaction picture Hamiltonian ℋI(t) corresponding to an atom-molecule interaction potential linear in the ladder operators J +, J -. The time evolution operators corresponding to several different approximations to ℋI(t) are constructed with Lie algebraic methods. We evaluate the vibrational transition probability as a function of the total collision energy, and also as a function of the strength of the Morse potential.
31 May 1991
TL;DR: The defining power of finite recursive specifications over the theory with + and · and λ (the state operator) over a finite set of states is found to be greater than that of the same theory without state operator.
Abstract: We investigate the defining power of finite recursive specifications over the theory with + (alternative composition) and · (sequential composition) and λ (the state operator) over a finite set of states, and find that it is greater than that of the same theory without state operator. Thus, adding the state operator is an essential extension of BPA (the theory of processes over +, ·). On the other hand, applying the state operator to a regular process again gives a regular process. As a limiting result in the other direction, we find that not all PA-processes (where also parallel composition λ is present) can be defined over BPA plus state operator.
TL;DR: The q-deformed differential calculus is proposed and analyzed in the framework of quantum plane as discussed by the authors, and the q-formed differential operator algebra is investigated and applied in the investigation of the quantum group SUq(2) and its representations.
Abstract: The q-deformed differential calculus is proposed and analysed in the framework of quantum plane The q-deformed differential operator algebra is investigated and applied in the investigation of the quantum group SUq(2) and its representations The generalization to n-dimensional differential calculus is made and shown to be a new solution to quantum plane by providing a new solution of the Yang-Baxter equation
01 Jan 1991
TL;DR: In this article, a quantum algebra A q (l, r) of order l and rank r transforms into a Lie algebra A(l,r) in the limit q→1.
Abstract: Quantum algebras were introduced at first in Refs.[1,2]. Then this concept was developed in details in Refs.[3,4] and in the papers of other authors (see for example [5–11] and the papers cited there). Because of deep analogy consisting between quantum and usual Lie algebras which is reflected in the fact that the quantum algebra A q (l,r) of order l and rank r transforms into usual Lie algebra A(l,r) in the limit q→1 a number of notations and theorems of the theory of Lie algebra representations can be transferred onto quantum algebras. In particular as it was shown in Refs [5–17] the q-analogs of well known quantities of Wigner-Racah algebra (WRA) (3j, 6j, 9j-symbols etc.) can be introduced. The detail investigation of the representation of quantum algebras was begun with the simplest quantum algebra SU q (2) that is a q-analog of the an-gular momentum theory (AMT) [18–21].
Posted Content•
TL;DR: The puncture operator in c=1 Liouville gravity is identified as the discrete state with spin J=1/2 and the recursion relation involving this operator is derived by the operator product expansion.
Abstract: We identify the puncture operator in c=1 Liouville gravity as the discrete state with spin J=1/2. The correlation functions involving this operator satisfy the recursion relation which is characteristic in topological gravity. We derive the recursion relation involving the puncture operator by the operator product expansion. Multiple point correlation functions are determined recursively from fewer point functions by this recursion relation.
TL;DR: The formalism developed in Part I of the present paper for the conjugate momentum of label operators is applied to standard physical examples, such as the regular one-dimensional lattice, the one dimensional box, and the angular momentum of the plane rotator.
Abstract: The formalism developed in Part I of the present paper for the conjugate momentum of label operators is applied to standard physical examples, such as the regular one-dimensional lattice, the one-dimensional box, and the angular momentum of the plane rotator. A non-standard application is also considered, i.e., the timeoperator, conjugate to the one-dimensional harmonic Hamiltonian.
TL;DR: In this article, a new linear transformation is introduced in order to construct isospectral potentials within the framework of supersymmetric quantum mechanics, where the adjoint character of the "ladder operator" is destroyed but with preservation of the energy spectrum.
Abstract: The authors introduce a new linear transformation in order to construct isospectral potentials within the framework of supersymmetric quantum mechanics. Under this transformation, the adjoint character of the 'ladder operator' is destroyed but with preservation of the energy spectrum leading to some interesting consequences such as the concept of pseudo-partnership of the two components. Connection with the Darboux Theorem as well as with some other recent methods of construction is analysed.
01 Jan 1991
TL;DR: $C^*$-algebras associated to two-step nilpotent groups by L. Baggett and J. Smith Representation of symplectic vector spaces obtained as unitary dilations by P. Jorgensen and D. R. Pitts.
TL;DR: In this article, a new method is proposed for finding the formal operator solution to the manifestly covariant canonical operator formalism of the four-dimensional quantum Einstein gravity without using any c-number background metric.
Abstract: A new method is proposed of finding the formal operator solution to the manifestly covariant canonical operator formalism of the four-dimensional quantum Einstein gravity without using any c-number background metric. The zeroth-order approximation of the gravitational field gpv is a nontrivial but manageable q-number. It is explicitly worked out how to construct the zeroth-order and first-order approximations to the operator solution.
TL;DR: In this paper, a method was proposed to express a product of SO(3)-tensor equivalents in terms of an operator basis which comprises all the equivalents, both diagonal and non-diagonal in angular momentum, multiplied by powers of the number of spins scalar n = 2 j.
TL;DR: An equation of motion for a Wigner operator in phase space is derived, which is the phase-space analog of the Heisenberg equation ofmotion for a quantum-mechanical operator and an application of this operator equation to time-dependent systems possessing invariants is considered.
Abstract: We have derived an equation of motion for a Wigner operator in phase space, which is the phase-space analog of the Heisenberg equation of motion for a quantum-mechanical operator. An application of this operator equation to time-dependent systems possessing invariants is considered, and the solution for the corresponding Wigner phase-space distribution function is obtained.
TL;DR: In this paper, the wave function is considered as the kernel of a many-particle operator, and the corresponding operator equation automatically leads to a convenient matrix algorithm for full configuration interaction calculations.
Abstract: We discuss modern trends in the theory and practice of full configuration interaction calculations. We pay the most attention to the wave operator method, in which the wave function is considered as the kernel of a many-particle operator. The corresponding operator equation, equivalent to the Schrodinger equation, automatically leads to a convenient matrix algorithm. We also discuss an alternative approach based on the pairing operator, generalizing the construction of the wave function in the method of one-particle spin-pairing amplitudes.
TL;DR: In this article, a linear continuous right inverse operator of a convolution operator in spaces of analytic functions with an exponential basis is given, which is similar to the one described in this paper.
Abstract: We give a description of a linear continuous right inverse operator of a convolution operator in spaces of analytic functions with an exponential basis.
01 Jan 1991
TL;DR: The great success of this theory provokes the question: "What convincing experiment contradicting it has ever been found?" No convincing experiment disproving it has yet to be found as discussed by the authors.
Abstract: Quantum theory is the most fruitful part of physics, no convincing experiment contradicting it has ever been found. The great success of this theory provokes the question:
TL;DR: In this paper, a finite Fourier series method for treating angular derivatives in the angular momentum term of the time-dependent Schrodinger equation in spherical coordinates is described, which involves a power series expansion of the evolution operator and treatment of singularities at φ = 0 by L'Hospital's rule.
TL;DR: In this paper, the authors transform the problem of solving the diffusion equation to that of solving a different linear differential equation involving a Hermitian operator H associated with the unperturbed system and an operator W=κ:M, which describes the effects of the velocity gradient κ°.
Abstract: For dilute solutions of polymeric molecules, the distribution function in the configuration space of a single molecule is determined by the so‐called ‘‘diffusion equation.’’ In the present discussion, we transform the problem of solving this diffusion equation to that of solving a different linear differential equation involving a Hermitian operator H associated with the unperturbed system and an operator W=κ:M, which describes the effects of the velocity gradient κ°. It is shown that the eigenvalues of H are the reciprocals of the time constants associated with the decay of perturbations of the system from equilibrium. It is then shown that the stress tensor may be written simply in terms of matrix elements of an operator M and a time‐development operator U with respect to the eigenfunctions of H as the bases. This development then leads to an expression for the relaxation modulus of linear viscoelasticity, which is of the form of the ‘‘generalized Maxwell model.’’ A formal expression for the relaxation...
01 Jan 1991
TL;DR: In this paper, a first-order differential operator that can serve as an alternative to the non-Euclidean wave operator to study SL(2,R) was derived.
Abstract: We derive a first-order differential operator that can serve as an alternative to the non-Euclidean wave operator to study SL(2,R)