Showing papers on "Ladder operator published in 1980"

Gelfand-Levitan method for operator fields

Abstract: The quantum generalization of the Gelfand-Levitan method is presented for the nonlinear Schroedinger model. The basic dispersion relation for operator Jost functions is derived, and the Heisenberg field operator is expressed in terms of scattering-data operators. Construction of Green's functions in the zero-density vacuum is discussed. The four-point function is explicitly calculated from the expression for the field operator and compared with the result of a direct Feynman graph summation. In addition it is proved for any number of particles that the Hamiltonian eigenstates constructed from the quantized scattering data are identical with those previously obtained by means of Bethe's ansatz.

The Ruelle-Araki transfer operator in classical statistical mechanics

Abstract: and resume.- The Kramers-Wannier transfer matrix.- The Ruelle-Araki transfer operator for one-dimensional classical systems.- Systems with long range interactions.- Zeta-functions of classical one-dimensional systems.

Coherent states for general potentials. IV. Three-dimensional systems

Abstract: The minimum-uncertainty coherent-states formalism is extended to higher-dimensional systems Specifically, for spherically symmetric three-dimensional potentials the formalism looks for coherent states which are products of an angular wave function times a radial wave function After reviewing the many studies on angular coherent states, I concentrate on the physically distinguishing radial coherent states The radial formalism is explained in detail and contrasted with the effective one-dimensional formalism The natural classical variables in the radial formalism are those which vary sinusoidally as g(E,L)theta(t), where theta(t) is the real azimuthal angular variable and g(E,L) is the number of oscillations between apsidal distances per classical orbit When changed to natural quantum operators, these operators can be given as the Hermitian sums and differences of the ''l'' raising and lowering operators The formalism is applied to the three-dimensional harmonic-oscillator and Coulomb problems

On an infinitesimal approach to semisimple Lie groups and raising and lowering operators of O(n) and U(n)

Abstract: A purely algebraic approach to the evaluation of the fundamental Wigner coefficients and reduced matrix elements of O(n) and U(n) is given. The method employs the explicit use of projection operators which may be constructed using the polynomial identities satisfied by the infinitesimal generators of the group. As an application of this technique, a certain set of raising and lowering operators for O(n) and U(n) are constructed. They are simpler in appearance than those previously constructed since they may be written in a compact product form. They are, moreover, Hermitian conjugates of one another, and therefore are easily normalized.

On necessary and sufficient conditions for the existence of time and entropy operators in quantum mechanics

Abstract: It is shown that time and entropy operators may exist as superoperators in the framework of the Liouville space provided that the Hamiltonian has an unbounded absolutely continuous spectrum. In this case the Liouville operator has uniform infinite multiplicity and thus the time operator may exist. A general proof of the Heisenberg uncertainty relation between time and energy is derived from the existence of this time operator.

Simple derivation of the Newton–Wigner position operator

Abstract: The kind of operator algebra familiar in ordinary quantum mechanics is used to show formally that in an irreducible unitary representation of the Poincare group for positive mass, the Newton–Wigner position operator is the only Hermitian operator with commuting components that transforms as a position operator should for translations, rotations, and time reversal and does not behave in a singular way that contradicts what can be learned from Lorentz transformations in the nonrelativistic limit.

Phase operator and phase fluctuations of spins

Abstract: It is shown that a Hermitian phase operator exists for quantum spins. Its spectrum is not continuous but has the values (2 pi /(2S+1))n, 0

Derivations on algebras of unbounded operators

Abstract: This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general *-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left EW#-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.

Raising and lowering operators and spectral structure - a concise algebraic technique

Abstract: We show that it is possible to give to any (linear and Hermitian) operator with discrete spectrum, acting on a Hilbert space, a canonical form in terms of the commutator and anticommutator of a pair of adequately constructed raising and lowering operators This construction enables one to investigate in a very compact way the (discrete) spectrum of the given operator, which becomes determined by a set of consistency conditions We apply the method to several important problems of quantum mechanics

Shift operator techniques for the classification of multipole‐phonon states. VI. Properties of nonscalar R(3) product operators in the R(2λ+1) groups

Abstract: Expressions connecting nonscalar R(3) products of operators which shift the eigenvalues of the R(3) Casimir operator L2 are constructed within the R(2λ+1) groups (λ=2 or 3).

Projection-operator approach to perturbation theory

Abstract: A perturbation theory is presented which is suitable for the treatment of strong or resonant interactions in quantum systems described by time-independent Hamiltonians. The formulation is exact for finite-level systems and encompasses both nondegenerate and degenerate problems. The derivation is based on the partitioning of the levelshift operator, an operator which occurs naturally through the use of projection operators. The formulation is applied to the eigenvalue problem and to the calculation of the transition amplitude between states of the unperturbed system induced by a time-independent perturbation. The results are expressed in terms of Green's functions involving continued fractions which are truncated for finite-level systems.

Shift‐operator techniques for the classification of multipole‐phonon states. II. Eigenvalues of the quadrupole shift operator O0l

Abstract: By the aid of previously derived relations involving shift operators and their products, a set of rules is set up for the determination of eigenvalues of the scalar Hermitian shift operator O0l in the space of R(5) nuclear quadrupole‐phonon states. The eigenvalues are listed for all seniority states with v<8.

The isotropic harmonic oscillator in an angular momemtum basis: An algebraic formulation

Abstract: A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harnomic oscillator, and L is its angular momentum vector. It is shown that H can be written in the form ℏω(2ν†ν + 놕λ + 3/2), where ν† and ν are raising and lowering (boson) operators for ν†ν, which has nonnegative integer eigenvalues k; and λ† and λ are raising and lowering operators for 놕λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form ℏω(2k + l + 3/2), previously obtained only by working in the coordinate or momentum representation. The common eigenvectors are constructed by applying the operators ν† and λ† to a "vacuum" vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν, ν†,λ, and λ†, in order to clarify their physical interpretation.

Spectrum of potentials gr-(s+2) via SL(2,R) acting on quaternions

Abstract: Potentials of the r-(s+2) power type are studied as the superposition of Yukawa potentials. The SL(s,R) group acting on the quadrivector impulsion considered as a quaternion generates a compact self-adjoint operator deduced from the Schrodinger operator by a Fourier-Fock transformation. The operator is approximated by finite rank operators and gives the spectrum of energy as a function of the coupling constant, the angular momentum and the exponent s.

The perturbed ladder operator method: closed form expressions of perturbed wavefunctions and matrix elements

Abstract: By mapping the perturbation scheme onto the ladder operator formalism, the field of application of the Schrodinger-Infeld-Hull factorisation method is enlarged. It is shown how, at each order of the perturbation, perturbed ladder operators can be constructed. Thus, without having to calculate explicitly either the excited unperturbed functions or any matrix element, one obtains analytical expressions of the perturbed eigenvalues in terms of the quantum numbers of the factorisable unperturbed problem. A three-terms recurrence relation, valid at any rank of the perturbation, is derived and leads to closed form expressions of the perturbed eigenfunctions. Consequently, a closed form expression of any matrix element on the basis of the perturbed eigenfunctions is easily obtained from the calculation of one unique particular integral.

Use of operator wave functions to construct a refined correspondence principle via the quantum mechanics of Wigner and Moyal

Abstract: We represent the momentum operator p by p+(h//2i)(∂/∂p) and the position operator q by q−(h//2i)(∂/∂p). We then apply each resulting operator that appears in the nonrelativistic quantum mechanics of Schrodinger and Heisenberg to the constant 1. What results is a Wigner–Moyal theory which is as complete as the usual Schrodinger–Heisenberg theory. Replacing Planck’s constant h/ by zero in the Wigner–Moyal theory results in the classical physics of Newton. We thereby obtain a refined correspondence principle.

Time evolution operator for N interacting quantum harmonic oscillators

Abstract: We discuss time evolution operator for N interacting quantum harmonic oscillators and determine certain conditions under which an explicit closed‐form expression can be found for this time‐ordered operator. The special case of N=2 is discussed, as an example.

Sturmian technique for the discrete spectrum of the Schrodinger equation

Abstract: Associates a compact self-adjoint operator A((-2mE)12/), parametrised by the energy E<0, with the Schrodinger operator H=p2/2m+V. By setting all eigenvalues of A equal to unity, the discrete spectrum of H is obtained formally. It is determined by means of the approximation of A using finite Hermitian matrices which are calculated in the so-called Coulomb Sturmian basis. The method is illustrated by a few examples previously reported (Yukawa potential, (-r- alpha ) potentials, 0< alpha <2, multiple centre potentials) and possible extensions are discussed.

The operators governing quantum fluctuations of Yang-Mills multi-instantons on S 4 and their Seeley coefficients

Abstract: We give explicit expressions for the Seeley coefficients of the fluctuation operator and the operator that appears in the Faddeev-Popov determinant, which arise in the calculation of quantum fluctuations around Yang-Mills multi-instantons.

From the Zwanzig-Feshbach projection method to the Landau-Zener approximation, a systematic time dependent approach

Abstract: Using the interaction picture, formal time dependent solutions of the Zwanzig-Feshbach projection operator method are derived which involve a nonunitary time-ordered memory operator. The familiar approximation which leads to a decoupling of the translational and internal motions in terms of a classical trajectory is derived by a systematic procedure which emphasises the generators of observable (operator) motion rather than the generators of wavefunction motion. Two-state approximations to this decoupled system are considered. These depend upon different approximations to the memory operator. Neglect of the memory operator together with the use of the stationary phase method produces the Landau-Zener approximation. Two extensions of this approximation are considered which depend upon different time disordered approximations to the two-state memory operator. Treating the continuum as a single state gives the bound-to-continuum analogues of these approximations.