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Showing papers on "Ladder operator published in 1976"


Journal ArticleDOI
TL;DR: In this article, the authors propose an unduly restricted framework for the theoretical description of quantum properties; non-hermitian operators, for instance unitary but also non-normal ones, may be acceptable as well if the projectors onto their eigenstates allow for a resolution of the identity operator, so as to preserve the probabilistic interpretation of the Hilbert space formalism.

161 citations


Journal ArticleDOI
TL;DR: In this paper, an analytically tractable approximation for the linearized Fokker-Planck collision operator describing a plasma nearly in thermal equilibrium was developed, which preserves the symmetry properties of the exact collision integral which imply the physical conservation laws, selfadjointness, and the H theorem.
Abstract: An analytically tractable approximation is developed for the linearized Fokker–Planck collision operator describing a plasma nearly in thermal equilibrium. This approximate operator preserves the symmetry properties of the exact collision integral which imply the physical conservation laws, self‐adjointness, and the H theorem. A renormalization procedure is developed to accurately treat collisions between particles of arbitrary masses. For large or small mass ratios, the approximate operator reduces to the standard expansions of the exact operator. In the case of identical particle collisions, the present approximation provides a significant improvement over the ’’model operator’’ previously given in the literature, yet retains the simplicity of former operators necessary for analytic work. The recalculation of the classical transport coefficients with this operator reduces to the solution of a coupled set of algebraic equations and indicates its reliability for use in complex neoclassical transport situations. The neoclassical electrical conductivity calculation demonstrates the new physical features of the approximate operator.

140 citations




Journal ArticleDOI
TL;DR: In this paper, a solution for the canonical commutation relation of the angular momentum operator and the phase was obtained for the problem of the phase phase phase decomposition problem, and a solution was also given for the case of the singularity.

23 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that an ultraweakly continuous completely dissipative operator on a W*-algebra A has a canonical form in terms of a completely positive map and a Hamiltonian provided that the cohomology groups of A with coefficients in a dual normal A-module are zero.
Abstract: It is proved that an ultraweakly continuous completely dissipative operator on a W*-algebra A has a canonical form in terms of a completely positive map and a Hamiltonian provided that the cohomology groups of A with coefficients in a dual normal A-module are zero.

23 citations


Journal ArticleDOI
TL;DR: In this article, the authors used character theory to obtain algebraic formulas for the 6j and 3jm symbols for SO3 ⊃ SO2, and the resulting calculation is somewhat simpler, algebraically speaking, than previous calculations and has the pedagogical advantage that only the concept of irreducible representation of a group is required, instead of the more elaborate concept of ladder operators.
Abstract: The various orthogonality and sum rules which the 6j and 3jm symbols satisfy are sufficient to obtain the algebraic formulas for these symbols for SO3 ⊃ SO2. Character theory enters in that the j's and m's occurring in the various sums are given by the triangle rule together with information on the symmetrized product and the branching, SO3 ⊃ SO2, The resulting calculation is somewhat simpler, algebraically speaking, than previous calculations and has the pedagogical advantage that only the concept of an irreducible representation of a group is required, instead of the more elaborate concept of ladder operators.

22 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived the conditions at the origin on eigenvectors of a nonrelativistic Hamiltonian, for the hydrogen atom or harmonic oscillator, and the condition that eigenfunctions of the angular momentum Lz must be single valued or continuous around a complete circle.
Abstract: Conditions on wave functions that are eigenvectors often have a mathematical rather than a physical basis; they are required simply because an eigenvector of an operator must be in the domain of that operator. This is illustrated with derivations of the conditions at the origin on eigenvectors of a nonrelativistic Hamiltonian, for example, for the hydrogen atom or harmonic oscillator, and the condition that eigenfunctions of the angular momentum Lz must be single valued or continuous around a complete circle.

17 citations


01 Jan 1976

13 citations


Journal ArticleDOI
TL;DR: In this paper, it was proved that the Bohm-Aharonov hypothesis concerning largely separated subsystems of composite quantum systems implies that it is impossible to express the dynamical evolution in terms of the density operator.
Abstract: It is proved that the Bohm-Aharonov hypothesis concerning largely separated subsystems of composite quantum systems implies that it is impossible to express the dynamical evolution in terms of the density operator.

13 citations


Journal ArticleDOI
TL;DR: In this article, the tensor product decomposition theorem is applied to the Thirring model and two types of expansions are obtained: strictly covariant expansions and semicovariant expansions.
Abstract: Operator products in quantum field theory on two-dimensional Minkowski space are expanded into a series of local operators by means of the tensor product decomposition theorem for representations of the conformal group. The Thirring model is used as an explicit example. Two types of expansions result. If the operator product acts on the vacuum state, we obtain strictly covariant expansions. In general however, each term in the expansion is only semicovariant.




Journal ArticleDOI
TL;DR: In this article, a theorem on the structure of weakly closed reductive operator algebras was proved, based on a known result of V. I. Lomonosov on transitive operators containing a nonzero compact operator.
Abstract: We prove a theorem on the structure of weakly closed reductive operator algebras. The proof essentially relies on a known result of V. I. Lomonosov on transitive operator algebras containing a nonzero compact operator. We deduce a number of corollaries which apply to the reductivity problem.Bibliography: 20 titles.

Journal ArticleDOI
TL;DR: The mathematical structures of the theories of a two-level atom interacting with radiation and an electromagnetic wave interacting with a dielectric medium in which it is propagated as a plane wave are identical as mentioned in this paper.
Abstract: The mathematical structures of the theories of a two-level atom interacting with radiation and an electromagnetic wave interacting with a dielectric medium in which it is propagated as a plane wave are identical. The spin structure of the Jones operator in optics representing the polarizer, is obtained in its general form in terms of Stokes parameters and identified with the optical density operator with a spin structure identical with the density operator in the quantum mechanics of the atom. While the quantum dynamical equation of the atom can be reduced to the gyroscopic form, the correspondence law giving expression to the parallelism of the two processes leads to forms of the dynamical law in topics which are identical with those of the quantum theory of the atom.

Journal ArticleDOI
TL;DR: In this article, Dirac bra and ket notation is developed for three-dimensional space and a number of transformations, closure relations, and projection operators are discussed in some detail for homogeneous isotropic and anisotropic systems, as well as to sums over photon polarization degrees of freedom.
Abstract: Dirac bra and ket notation is developed for three‐dimensional space. A number of transformations, closure relations, and projection operators are discussed in some detail. The formal results are applied to examples pertaining to homogeneous isotropic and anisotropic systems, as well as to sums over photon polarization degrees of freedom. Most of the discussion is tailored for the student before his exposure to operator methods of quantum mechanics.


Journal ArticleDOI
01 Jan 1976
TL;DR: A characterization of the operators A for which the equation TX XV = A is solvable is given in this article, where T is a fixed right invertible operator and V a fixed unilateral shift.
Abstract: A characterization of the operators A for which the equation TX XV = A is solvable is given, where T is a fixed right invertible operator and V is a fixed unilateral shift. The aim of this note is to give a characterization of the solutions of the equation TX XV = A, where T, V, A are given operators acting in a Hilbert space, and V is a unilateral shift. As a by-product we also give a sufficient condition for A to be expressed in the form A = VX XV = V* Y YV* = V*Z ZV with V a unilateral shift. The problems studied throughout the paper originate from a question of C. Foia§. The author expresses his gratitude to Bernard Morrel for pointing out some errors in the manuscript. Let H be a complex Hilbert space and denote by L(H) the algebra of all bounded linear operators acting in H. For any T, S, A G t(H) put dn(A;T,S) = 2 TjASk, n > 0. j+k=n It is easy to see what we have Tdn{A;T,S)-dn{A;T,S)S = d„(TA -AS;T,S) = Tn+1A -ASn+l, dn+l(A; T,S) = Tdn(A; T,S) + AS"+] = d„(A; T,S)S + T"+lA. 1. Proposition. Let B G £(//) and let V be a unilateral shift such that TB BV = 0, B(I VV*) = 0. Then we have 5 = 0. Proof. By induction we derive BV" = T"B, so BV"(I VV*) = 0, n > 0, and this implies 5 = 0. Received by the editors November 6, 1974 and, in revised form, August 20, 1975. A MS (MOS) subject classifications (1970). Primary 47A50; Secondary 47B47.

Journal ArticleDOI
TL;DR: In this paper, Dirac's perturbation solution of the time-dependent Schrodinger equation is formulated by means of a well-defined operator PD with the properties ======\/\/\/\/\/\/£££ £££€££$££• ££€ ££ £€£ £ ££•££
Abstract: Dirac's perturbation solution of the time-dependent Schrodinger equation ih, is formulated by means of a well-defined operator PD with the properties where ϕn(t)〉 is the n-th order correction of ϕ(t)〉. This operator PD can be used to solve the equation of motion of the evolution operator, of the statistical operator, and of quantum mechanical conserved quantities in a corresponding way. A simple proof of the quantum mechanical adiabatic theorem shows this formulation of perturbation corrections to be useful.

Journal ArticleDOI
TL;DR: In this paper, an angular momentum operator identity which was previously established using matrix representations was re-derived using Feynman's operator differential equation technique, and the identity was used to obtain the angular momentum.
Abstract: An angular momentum operator identity which was previously established using matrix representations is re-derived using Feynman's operator differential equation technique.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a generalized form of idempotency of Dirac's density operator is a necessary and sufficient condition for semigroup property of Bloch's density operation.

Journal ArticleDOI
TL;DR: In this paper, the polarization operator of the phonon Green's function of the Frohlich Hamiltonian was calculated in the first approximation, which corresponds to the assumption that the electron momenta are orthogonal to phonon momentum.
Abstract: We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.


Journal ArticleDOI
TL;DR: In this paper, the authors derived a stability criterion for linear multivariable systems by using an operator perturbation technique to try to ensure that the operator in the transformed system equation is a contraction.
Abstract: By invoking a fixed-point theorem of functional analysis. Freeman has developed a stability criterion for linear multivariable systems. The criterion is derived by using an operator perturbation technique to try to ensure that the operator in the transformed system equation is a contraction. It is clearly desirable to be able to choose the perturbation operator so that the transformed open-loop operator is as small as possible in some sense. In the letter, we determine that perturbation operator, from the class of diagonal linear operators, that gives the smallest contraction for the transformed open-loop system operator.

Journal ArticleDOI
TL;DR: In this article, a new approximation operator is introduced and its properties are studied by using probabilistic tools such as the Chebishev inequality and Liapounov's central limit theorem.
Abstract: In this paper a new approximation operator is introduced and its properties are studied. Special cases of this operator are the well-known Szasz power-series approximation operator and its generalization by D. Leviatan. The behaviour of the new approximation operator at points of continuity and discontinuity is investigated by using probabilistic tools as the Chebishev inequality and Liapounov’s central limit theorem. Such probabilistic methods of proof simplify the proofs and give better understanding of the approximation mechanism.