Showing papers on "Ladder operator published in 1973"

On the theory of brownian motion

Abstract: The spectrum of the Fokker-Planck operator for weakly coupled gases is considered. The operator is decomposed into operators acting on functions whose angular dependence is given by spherical harmonics. It is shown that the operator corresponding to l = 0 has zero for a point eigenvalue (the eigenfunction is the Maxwell distribution). There are no other point eigenvalues and the continuous spectrum of all of the operators is the entire negative real axis. Some consequences are briefly discussed.

Off-diagonal operator equivalents

Abstract: Rules for constructing operator equivalents off-diagonal in the angular momentum quantum number are described. Hyperbolic angular momentum operators are used in conjunction with the usual (spherical) angular momentum operators to obtain equivalents for all values of Δj and Δm. Tables of operator equivalents for harmonics of rank 0–6 are given.

Quantization in generalized coordinates III-Lagrangian formulation

Abstract: It is shown that if one incorporates the generalized coordinate quantum velocitiesQ1 as given byQ1=l[H,Q1](h=1) into the generalized classical Lagrangian for a free particle (the total energy),L=1/2Q1gtkQk one does not obtain (no matter what ordering of the operatorsq l ,q k andglkwe choose the correct quantum Lagrangian operator which is a transformation from -1/2V2 to generalized coordinates (Gruber, 1971, 1972).ql as given byql=i[H,ql] turns out to be the Hermitian part of a more generaiized operator which we call the total generalized velocity operator similar to the notation in ear previous articles (Gruber, 1971, 1972). This total velocity operator really determines the fundamental structure governing our system in the Lagrangian formulation. We show that ft is through the total velocity operator that we make the transition from classical to quantum mechanics and through our procedure we arrive at the correct quantum Lagrangian operator.

Position operators in relativistic quantum theory: Analysis of algebraic operator relationships

Abstract: This work is the first of a series of three papers examining different aspects of position operators in relativistic quantum theory. In this paper the properties of the position operator X and the velocity operator V are derived for single particle matrix elements in the context of the Poincare generator algebra. Both the physical meaning and the mathematical implications of each property are discussed. The algebraic structure of the extended set of relationships including the Poincare generators, X and V is examined. It is found that this set defines an infinite algebra which is intractable mathematically. The Casimir operators of the Poincare algebra are required to be Casimir operators for X and V, a new condition on V is formulated, and a simple solution for K is constructed. These conditions, together with familiar position operator properties, give the constraints and solutions for the extended algebra.

Matrix elements of off-diagonal operator equivalents

Abstract: A simple derivation of off-diagonal operator equivalents and their matrix elements is given in the framework of Schwinger's coupled boson representation of angular momentum theory. The operator equivalents are up to a factor identical with those recently given by Atkins and Seymour, but they are derived differently without making use of properties of spherical harmonics, symmetrization, 3j coefficients and the Wigner-Eckart theorem.

A new position operator for the photon

Abstract: We show how to construct a position operator for the photon in its own reference system. This operator is Hermitian and has commuting components.

Integralproperties of hermite polynomials by operator methods

Abstract: The quantum mechanical method of the occupation number representation of the harmonic oscillator and the technique of noncommuting operators is used in the derivation of a master integral for Hermite polynomials. It contains not only nearly all integrals of this type given in the literature but gives more than twice this number new ones. The advantage in the derivation of the master integral is that it needs only elementary operator commutations.

On Operator Equations in Particle Physics

Abstract: 1 Many problems in Particle Physics and Quantum Field Theory reduces to the problem of finding solutions of certain operator equations The most popular are operator equations for currents, eg, $$\left[ {J_\ell (x),\,J_k (y)} \right] = iC_{2k}^s \delta _3 \left( {\overline x - \overline y } \right)J_s (\overline x )$$ (1) or dynamical equations for scalar quantum fields like $$( + m^2 )\Phi (x) = \lambda \Phi ^n (x)$$ (2) The commonly used method of solutions of operator equations like (1) is based on Weyl trick which he used for a classification of representations of Heisenberg commutation relations It consists of the association with the commutation relations (1), the well-defined infinite-dimensional topological group G Using then the well elaborated global representation theory of G and passing to the infinitesimal representations, one obtains a class of solutions of current commutation relations (1) (cf Reference 1)

Discrete-continuous hydrogenic r/sup k/ integrals by a ladder-operator procedure

Abstract: It is shown that the accelerated'' or multistep'' ladder-operator procedure, when introducing pseudo-key matrix elements, leads to a closed-form expressions for the discrete-continuous hydrogenic r/sup k/ matrix elements. (auth)

A note on Kato's perturbation theory

Abstract: The results of T. Kato are expanded by generalizing the relative bound condition on the perturbation to determine the domain of powers of the perturbed operator and by exhibiting some useful relative bounds between the unperturbed operator, the perturbed operator, and the perturbation.

Nondegenerate operator algebras in spaces with an indefinite metric

Abstract: A theorem on the bicommutant is proved for nondegenerate -symmetric operator algebras in the space . By means of this theorem a simple description is given of the set of unitarily equivalent classes of nondegenerate, weakly closed -symmetric algebras.

On the semiboundedness of the (ø4)2 Hamiltonian

Abstract: An elementary alternate proof of the semiboundedness of the locally correct HamiltonianH0+ί:o4(x):g(x)dx of the (o4)2 quantum field theory model. The interaction operator is expressed as the sum of a positive operator and operators which are “tiny” relative to LNe for any ɛ>0, whereN is the number operator.

Continuity of linear fractional transformations on an operator algebra

Abstract: An operator coefficient linear fractional automorphism SF on the unit ball of operators is continuous in the weak operator topology if and only if ¡F(0) is compact. Let SS denote the set of bounded operators on the Hubert space H which have norm not greater than one. A map F:38—>3$ of the form (1) F(J) = (C + DJ)(A + BJ)-1 for each J e 3§ is called general symplectic when A, B, C, and D are operators on H which satisfy (2) AA* BB* = I = DD* CC*, (3) AC* = BD*. Fixed point theorems for general symplectic maps are of particular interest. The best proof of the main fixed point theorem (Pontryagin-H. Langer) for these maps is an application of the Schauder-Tychonoff theorem and is due to M. G. Krein [1]. His proof consists of showing that if F is a general symplectic map of form (1) where the operators B and C are compact operators, then ¡F is continuous in the weak operator topology (abbreviated w.o.t.). In this note, we prove that the converse is true, namely Proposition. If F is continuous in the weak operator topology then B and C are compact operators. Proof. The following facts will be useful: Equation (2) implies that A and D are invertible. Set S=A~1B. Equation (3) implies that S* = D^C. Equations (2) and (3) combined imply ||S||<1; consequently l—S*S is invertible. Received by the editors May 12, 1972 and, in revised form, August 7, 1972. AMS (MOS) subject classifications (1970). Primary 47B50, 74H10. 1 This result is contained in the author's doctoral dissertation written at Stanford University. The author was supported by an NSF graduate fellowship. © American Mathematical Society 1973 217 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

Problem of the time operator and collision duration

Abstract: The notion of collision duration and properties of the symmetrical time operator in the nonrelativistic quantum mechanics are investigated. Using the theory of genenalized extensions of symmetrical operators, a conclusion is made: not only self-adjoint but also maximal symmetrical operators can correspond to the observables in the quantum mechanics. (auth)