Showing papers on "Operator (computer programming) published in 1969"

An Exact Quantum Theory of the Time‐Dependent Harmonic Oscillator and of a Charged Particle in a Time‐Dependent Electromagnetic Field

Abstract: The theory of explicitly time‐dependent invariants is developed for quantum systems whose Hamiltonians are explicitly time dependent. The central feature of the discussion is the derivation of a simple relation between eigenstates of such an invariant and solutions of the Schrodinger equation. As a specific well‐posed application of the general theory, the case of a general Hamiltonian which settles into constant operators in the sufficiently remote past and future is treated and, in particular, the transition amplitude connecting any initial state in the remote past to any final state in the remote future is calculated in terms of eigenstates of the invariant. Two special physical systems are treated in detail: an arbitrarily time‐dependent harmonic oscillator and a charged particle moving in the classical, axially symmetric electromagnetic field consisting of an arbitrarily time‐dependent, uniform magnetic field, the associated induced electric field, and the electric field due to an arbitrarily time‐dependent uniform charge distribution. A class of explicitly time‐dependent invariants is derived for both of these systems, and the eigenvalues and eigenstates of the invariants are calculated explicitly by operator methods. The explicit connection between these eigenstates and solutions of the Schrodinger equation is also calculated. The results for the oscillator are used to obtain explicit formulas for the transition amplitude. The usual sudden and adiabatic approximations are deduced as limiting cases of the exact formulas.

Ordered Expansions in Boson Amplitude Operators

Abstract: The expansion of operators as ordered power series in the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required $c$-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,\ensuremath{-}1$, respectively, of an order parameter $s$. In terms of this convention it is shown that for bounded operators the coefficients are finite when $sg0$, and the series are convergent when $sg\frac{1}{2}$. For each value of the order parameter $s$, a correspondence between operators and $c$-number functions is defined. Each correspondence is one-to-one and has the property that the function $f(\ensuremath{\alpha})$ associated with a given operator $F$ is the one which results when the operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the $P$ representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter $s$.

The Human as an Optimal Controller and Information Processor

Abstract: A mathematical model of the instrument-monitoring behavior of the human operator is developed. The model is based on the assumption that the operator behaves as an optimal controller and information processor, subject to his inherent physical limitations. The resulting model depends explicitly on the control task and the control actions. Provision is made for the ability to obtain information from the peripheral visual field. There are no restrictions on signal coupling. The specific characteristics of the operator's visual sampling behavior are predicted by solving a nonlinear, deterministic optimization problem. A two-axis compensatory tracking example is investigated, and the results exhibit the general characteristics expected of a human operator performing a similar task.

Discrete space theory of radiative transfer I. Fundamentals

Abstract: This paper sets out a formal theory of radiative transfer in one-dimensional scattering media of arbitrary physical constitution. The theory is based on an extension of the treatment of Redheffer, in which the response of a layer of arbitrary thickness to fluxes incident on its boundaries is described by a certain linear operator. Juxtaposition of two such layers gives a third layer, whose operator can be related to those of its constituents by an operation designated as the star product. It is shown that this set of operators constitutes a semigroup under the star product, and that the infinitesimal generators of the semigroup can be computed in term s of the physical properties of the medium , point by point. This makes it possible to write equivalent discrete and differential equations from both of which transmission and reflexion operators, the emission due to internal sources, and the internal fluxes at prescribed levels in the medium can be obtained.

Generalized Bose Operators in the Fock Space of a Single Bose Operator

Abstract: Generalized Bose operators b which reduce by two the number of quanta of a Bose operator a are studied in the Fock space of a. All representations of the b's as normal‐ordered (infinite degree) power series of the a's are found. The unitary operators relating the irreducible components of b to a are also exhibited. The analogous result for b(k)'s which reduce the number of a quanta by k is given and the limit k → ∞ is discussed.

Some bilinear convergence characteristics of the solutions of dissymmetric secular equations

Abstract: If an approximate eigenfunction is obtained by the solution of a dissymmetric set of secular equations formed by the use of different expansion functions to precede and follow the operator, it is shown here that the error in the eigenvalue is proportional to $\mu^\dagger\mu$ where $\mu^\dagger$ and $\mu$ are measures of the amounts by which the pre- and post-expansion functions are not able to fit the adjoint and direct eigenfunctions of the operator This replaces the $\mu^2$ error in Rayleigh-Ritz variation theory This result is of considerable value for the circumstances where the use of different sets makes the integrals evaluable for specially desirable post-expansion functions The introduction of direct electronic correlation into wavefunctions is a case where the integrals can be evaluated with different sets of functions but not with the same set Further, these results show how a particular use of numerical integration gives eigenvalues with errors of lower order than those associated with the same integration procedure in normal integrals

Coupling Operator Method in the SCF Theory

Abstract: The coupling operator method in the general SCF theory is discussed. It is shown that there is an abritrariness in the definition of the general SCF operators. In the course of discussion, several SCF equations useful in practical applications are presented.

Theory of the Influence of Environment on the Angular Distribution of Nuclear Radiation

Abstract: The effect of electronic relaxation processes on the angular correlation and on the angular distribution of radiation from oriented nuclei is investigated. The influence of the environment on the radioactive nuclei is taken into account by reducing the density operator for the total system (nucleus and surroundings mutually interacting) to a density operator for the nucleus alone. Elimination of the unobserved bath variables is performed with the help of Zwanzig's projection-operator technique. The Liouville formalism is used throughout. The (initially unspecified) properties of the environment enter the theory via second-order correlation functions, which are defined in terms of equilibrium ensemble averages of certain bath operators, like, e.g., the hyperfine-field operator. The matrix elements of the nuclear-evolution operator (which is a superoperator in Liouville space) with respect to a complete orthonormal set of multipole operators are just the usual perturbation factors ${{G}_{k{k}^{\ensuremath{'}}}}^{q{q}^{\ensuremath{'}}}$ of perturbed-angular-correlations theory. The consequent use of the multipole representation yields immediately the final formulas needed in the expression for both the angular distribution of radiation from oriented nuclei and the angular correlation function. The general theory includes relaxation processes due to magnetic and quadrupole interactions. The important case of purely magnetic interactions is discussed in more detail. Specialization to relaxation caused by randomly fluctuating fields yields a formula which contains both the Abragam-Pound result for time-fluctuating quadrupole interaction and Micha's extension to randomly time-varying magnetic fields in multidomain ferromagnets. Exact high-temperature solutions are presented for single crystals in a static magnetic field and with magnetic-type relaxation processes (axially symmetric case). For nuclei with spin $I=1$, the extension to arbitrary temperatures has been considered. The application of the present theory to the problem of multipole relaxation (which arises, e.g., in spin-lattice relaxation measurements with NMR/ON technique) is discussed.

Notes towards the construction of nonlinear relativistic quantum fields. III: Properties of the $C^*$-dynamics for a certain class of interactions

Abstract: 1. This note treats the C*-dynamics of positive-energy symmetric (or 'Bose-Einstein') quantum fields in continuation of [ l ] . The temporal development of the systems being considered is given by a one-parameter group of automorphisms of a C*-algebra, which in general are not unitarily implemented, but may by a process of localization be reduced to the consideration of a complex of putative one-parameter unitary groups. Each such group is to be generated by an operator H' which is formally given as H+ V, where each of H and V may be formulated as a selfadjoint operator in Hubert space, but whose sum is a priori ill-defined as such because of the singular nature of V in relation to H. In [l , I ] a theory of renormalized products of quantum fields was initiated which served as a basis for the treatment of the operators V of concrete interest. I t followed that for a certain class of relativistic cases: (a) H+V is densely defined and has a selfadjoint extension H'\ (b) the associated complex of one-parameter unitary groups corresponds to a C*-automorphism group provided the Lie formula: e' =limn(e e) is applicable (as is the case e.g. if H' is unique, by a theorem of Trotter). In the present note, by making a natural use of mild particularities of the operators in question, a selfadjoint extension H' is constructed which has the modified property, sufficient for the construction of an appropriate C*-automorphism group, that e' = limmlimn(e e), if {fm} is any sequence of real functions of compact support on R such that/m(X)—>X and |/m(X)| ^ | X | ; and this operator has in addition many other relevant properties. The treatment is quite general, and apart from the finiteness of the moments of V and e~, and the nonvanishing of the 'mass/ makes no significant assumptions.

Toward a Theory of Enumerations

Abstract: An attempt is made to show that there is much work in pure recursion theory which implicitly treats computational complexity of algorithmic devices which enumerate sets. The emphasis is on obtaining results which are independent of the particular model one uses for the enumeration technique and which can be obtained easily from known results and known proofs in pure recursion theory.First, it is shown that it is usually impossible to define operators on sets by examining the structure of the enumerating devices unless the same operator can be defined merely by examining the behavior of the devices. However, an example is given of an operator which can be defined by examining the structure but which cannot be obtained merely by examining the behavior.Next, an example is given of a set which cannot be enumerated quickly because there is no way of quickly obtaining large parts of it (perhaps with extraneous elements). By way of contrast, sets are constructed whose elements can be obtained rapidly in conjunction with the enumeration of a second set, but which themselves cannot be enumerated rapidly because there is no easy way to eliminate the members of the second set.Finally, it is shown how some of the elementary parts of the Hartmanis-Stearns theory can be obtained in a general setting.

Numerical Operators for Statistical PERT Critical Path Analysis

Abstract: This paper extends the algorithm developed by Hartley and Wortham [2] for the calculation of the c.d.f. of the critical time in a PERT network. Additional first order crossed networks and their operators are given. In addition, a technique for developing an integral operator for any PERT network is presented.

Absolute continuity of hamiltonian operators with repulsive potential

Abstract: Introduction. One expects that the absolutely continuous part of the spectrum of a Hamiltonian operator H= -A+ V in L2(En) (where A is the Laplacian operator and V is the operation of multiplication by a real function which approaches 0 at oo) will be the interval [0, oo). That this is the essential spectrum has been shown under very weak assumptions on V [7], but the absolute continuity has been demonstrated only under much stronger assumptions [l], [2], [3], [8]. In this paper we prove that for smooth positive potentials V which are sufficiently repulsive outside some bounded set, the operator -A+ V is absolutely continuous. Our conditions are similar to those in the previous work of Odeh [5]. We use results of Putnam [6] on commutators of pairs of selfadjoint operators. Our method works for dimensions n= 1, 2, or 3, though we consider only two cases, n= 1 (because of its simplicity) and n =3 (because of its importance for applications). Only partial results seem possible in higher dimensions.

Universal operators and invariant subspaces

Abstract: For any Banach space X, let B(X) denote the space of continuous endomorphisms of X. An operator U in B(X) will be called universal if, given any T in B(X), then some nonzero multiple of T is similar to a part of U i.e. there exists XEC, X 0, a closed subspace Xo of X such that UX0CXo and a linear homeomorphism q of X onto Xo such thatXT=q-'(Uj Xo)q. The first example of a universal operator (or model) was constructed by G.-C. Rota [1] for the Hilbert space case. In that instance, U is (unitarily equivalent to) the direct sum of countably many copies of the reverse shift , ( 2, 23, * * * ) > (2i 4, 4, * . . ). Such a direct sum obviously defines an operator whose nullspace is infinite-dimensional and whose range is the whole space. In this note, we show that all such operators are universal (when X is a separable Hilbert space) and that, with rather obvious modifications, the arguments extend to arbitrary Banach spaces.

Projected Hartree Product Wavefunctions. II. General Considerations of Young Operators

Abstract: A discussion is given of the forms of two specific and one general Young operator for the irreducible representations of Sn important for fermion space functions. Comparisons are made of the projected Hartree product version of Lowdon's projected Hartree–Fock method with CI calculations.

Theory of Reactive Scattering. I. Homogeneous Integral Solution Formalism for the Rearrangement τ‐Operator Integral Equation

Abstract: The integral equation for the rearrangement τ operator is used as the basis for a discussion of reactive scattering. The concept of the amplitude density introduced by Johnson and Secrest is extended to reactive collisions by means of the τ operator. The homogeneous integral solution method of Sams and Kouri is used to develop a noniterative solution to the integral equation satisfied by the reactive‐scattering amplitude density. This method is characterized by very stable behavior and is capable of quite accurate solutions with a relatively large step size. We consider the formulation of the method first for a general reactive collision. Both open and closed channel contributions to the reactive scattering amplitude density are considered. We employ the Feshbach projection operator formalism to treat the effects of complex formation due to virtual excitation of internal degrees of freedom. After discussing the general reactive collision, we next deal with the specific problem of atom–diatom reactions. Notation and techniques developed by Miller are used to treat problems associated with coordinates. Unlike Miller's discussion of reactive scattering, the present approach does not require inclusion of square integrable functions since the integral equation for the τ operator (and the equation for the amplitude density) already takes explicit account of the effects described by such terms. Further, the present approach allows one to easily study the effects of vibrational and/or rotational excitation on the reaction rate since the initial state appears explicitly as an inhomogeneity in the integral equation for the reactive scattering amplitude density. Finally, we discuss the manner in which the T matrix may be obtained from the reactive scattering amplitude density. It results that the reactive T matrix may be expressed in terms of the nonreactive T matrix.

Operator guidance and control termianl

Abstract: A display and control terminal which in response to operator initiation is utilized to display operator guidance information and control the insertion of data in conformity with the displayed guidance information.

Nonunitary operators with absolutely continuous spectrum

Abstract: Sufficient conditions are found for the linear similarity of a nonunitary operator T to a unitary operator with absolutely continuous spectrum. The results obtained are applied to second-order differential operators.

Correspondence identities. iv. the rutherford scattering identity.

Abstract: A complete correspondence identity is obtained for the electron-proton system, whereby the non-relativistic quantum dynamics of the system is obtained from solutions of the corresponding classical problem and their analytic continuation given by the authors in the previous paper. The kernels of the spectral operator IE = δ(E - H) in momentum and symmetric representation are obtained as sums over classical action functions for all non-zero real energies E. A general derivation of a scattering cross section from a spectral operator is presented, and applied to this system: the long-range distortion appears naturally. By this means and alternatively in terms of transition operators it is shown how the correct quantum-mechanical differential scattering formula follows from classical Rutherford theory. Complete correspondence identities are discussed. Quantum-mechanical barrier penetration is obtained through analytic continuation of classical action functions. A model of the system based on classical electron orbits is an improvement on the Bohr-Sommerfeld model.

The operator gap

Abstract: This paper continues investigations pertaining to recursive bounds on computing resources (such as time or memory) and the amount by which these bounds must be increased if new computations are to occur within the new bound The paper proves that no recursive operator can increase every recursive bound enough to reach new computations In other words, given any general recursive operator F[ ], there is an arbitarily large recursive t( ) such that between bound t( ) and bound F[t( )]( ) there is a gap in which no new computation runs This demonstrates that the gap phenomenon first discovered by Borodin for composition is a deeply intrinsic property of computational complexity measures Moreover, the Operator Gap Theorem proved here is shown to be the strongest possible gap theorem for general recursive operators The proof involves a priority argument but is sufficiently self-contained that it can easily be read by a wide audience The paper also discusses interesting connections between the Operator Gap Theorem and McCreight & Meyer's important result that every complexity class can be named by a function from a measured set

The groupF4 and its generators

Abstract: The representations ofF4 are discussed and the decompositions of some representations in the reductionsSO26 →F4 →SO9 are tabulated. The decompositions of some inner products ofF4 are also tabulated. A general method of constructing operator bases for representations of larger groups is given. Using this method, operators corresponding to the generators ofF4 acting on the representation (1000) are constructed. This method also generates relations between 6-j symbols and some examples are given.

On Proper Time and Localization for the Quantum Relativistic Electron

Abstract: position operators (which are generally considered as covariant) are studied in regard to their behaviour under the inhomogeneous Lorentz group and under ordinary and proper time evolution. They are Pryce's class d, Bacry's and R. J. Finkelstein's operators. It results that all three have some unreasonable properties for Dirac's electron. The reason for the difficulty in the case of Finkelstein's operator can be traced back to an incorrect definition of it. By making some changes a satisfactory position operator is obtained but it turns out to be equivalent to Bunge's one (up to an arbitrary and unimportant constant of motion). The results of this paper reinforce those of previous work which show that (for Dirac's electron) Bunge's position operator seems to be in a privileged place as regards the solution of the localization problem. Incidentaly, the interpretation of some relations previously used as auxiliary formulas for computations, allows us to obtain a manifestly covariant expression for Dirac's Hamiltonian and a second manifestly covariant form of Dirac's equation.

On the potential operator for one-dimensional recurrent random walks.

Abstract: where p(n) denotes the «-fold convolution of xt with itself. Let fi(8), 8 e Rd, denote the characteristic function of ti. Let X denote the closed subgroup of Rd generated by the support of /x. Without loss of generality, we can assume that X is rf-dimensional. Then d= 1 or d=2. If d= 1, let a2 denote the variance of /x. In [3] S. Port and the author showed that there is always a potential operator associated with xt that has useful properties. In the two-dimensional case our results were self-contained. In the one-dimensional case, however, the proof consisted of a reduction to the nonsingular case which had been treated earlier by Ornstein [1] and [2]. All of these results are, of course, extensions of results of Spitzer (see [5]) for lattice random walks. The main purpose of this paper is to present a new and simpler proof of those consequences of Ornstein's work which were used in [3]. The result of [3] which depended on Ornstein's work will be stated here as

Apparatus for working under water

Abstract: APPARATUS FOR WORKING UNDER WATER COMPRISING A COMBINATION OF A SUBMERGIBLE WORKING MACHINE SUCH AS A BULLDOZER AND A FLOATING BODY EQUIPPED WITH VARIOUS DEVICES NECESSARY FOR OPERATING THE WORKING MACHINE. A SUBMERGIBLE CONTROL CHAMBER ACCOMMODATING AN OPERATOR IS SUSPENDED INTO WATER FROM THE FLOATING BODY TO ENABLE THE OPERATOR TO CONTROL THE OPERATION OF THE WORKING MACHINE.