Time-independent Methods Research Articles

Unification of polaron and soliton theories of exciton transport.

We present a unified theory of polaron and soliton dynamics by combining time-dependent variational methods recently applied to the theory of Davydov solitons with partial-dressing methods well known from polaron theory. We focus on the simplest partial-dressing assumption, applying a common dressing fraction to all phonon modes. Our fundamental result is a system of nonlinear evolution equations in which the tendency of a system to form Davydov solitons is balanced against its tendency to form small polarons. We subsequently apply time-independent variational methods to determine the optimal dressing fraction in a mean-field manner. The characterization of the partially dressed soliton states that results is complete with respect to the system parameter space. Consistent with prior works from polaron theory, we find a self-trapping transition that is only weakly modified by our inclusion of nonlinearities, and we reinterpret this transition in terms of our newly obtained soliton states. Applying our results to a central problem in bioenergetics, we obtain results markedly different from well-known results of Davydov's theory.

Incorporating advantages of time-dependent dynamics in time-independent collision methods: Early asymptotic analysis

Transition amplitudes that may be written as matrix elements of a Green’s operator are quite usefully cast as temporal correlation function of localized wave packets. In many situations, the correlation function is nonzero only for a short time while the wave packet is close to its initial position. The transition amplitude is only sensitive to the potential near the initial location of the wave packet. We derive time-independent analogs of the above described features of time-dependent collision theory for matrix elments of the time-independent Green’s function. Transition amplitudes are shown to be completely independent of the potential outside the Franck–Condon region when the true wave function can be approximated by a (primitive or uniform) semiclassical form in the outer region. The reaction coordinate can be separated into a strongly interacting Franck–Condon region, to be treated by standard close coupling methods, and an asymptotic region for which no dynamical calculations are required in the semiclassical limit. Because this is a spatial, rather than temporal, separation, our semiclassical method is successful in situations, such as predissociation, where time-dependent methods fail. Also, we can evaluate total cross sections by using basis sets that are optimal for the Franck–Condon region without ever considering the transition to asymptotic target states. Numerical illustrations are provided for predissociations in Na2 and Br2.

Assessment of Interaction Among Three Carcinogens on Rat Mammary Carcinogenesis in a Factorially Designed Experiment<xref ref-type="fn" rid="FN2">2</xref>

Female Sprague-Dawley rats were given by stomach tube 7,12-dimethylbenz[a]anthracene [(DMBA) CAS: 57-97-6] on the 77th day of age at the rate of 1.6 mg/100 g body weight, or procarbazine [(PCZ) CAS: 671-16-9] on the 84th day of age at the rate of 10 mg/100 g body weight, or 0.5 Gy of total-body x-rays on the 91st day of age, singly or in all possible combinations or no treatment. All rats were studied for mammary carcinogenesis for 370 days after the 84th day of age. Three measures of mammary carcinogenesis were studied. These were the incidence of rats with mammary adenocarcinomas, or mammary fibroadenomas, or mammary neoplasia of either type. Each of these measures was studied also for rats with 2 or more or 3 or more mammary neoplasms. Assessment of possible interaction among the three carcinogens with regard to the incidence of neoplasms was done by time-independent or time-dependent methods, both of which gave remarkably consistent results. For rats with 1 or more adenocarcinomas, 1 or more fibroadenomas, or 1 or more adenocarcinomas and/or fibroadenomas, both methods showed no interaction among the carcinogens, which can, therefore, be considered to have produced additive effects. An exception to this finding of additivity was an apparent synergistic interaction between DMBA and PCZ when the measures of rats with 2 or more, or 3 or more mammary neoplasms of either type, or 3 or more fibroadenomas were analyzed; these analyses, however, were based on relatively small numbers of rats with multiple tumors. Since no interactions were found for the usual measure of carcinogenesis, namely, incidence of rats with 1 or more neoplasms, the overall conclusion is that DMBA, PCZ, and x-ray act additively in the induction of mammary neoplasms in the female Sprague-Dawley rat.

Methods for solving inverse problems for radiation transport—an update

Abstract Recent analytical results of both time-dependent and time-independent methods are surveyed for inferring the albedo of single scattering and the coefficients of the Legendre expansion of the angle-dependent scattering law. Also presented are some new equations and new approximate solution techniques for the time-independent method that may help reduce the poor conditioning of the numerical solution of the formal ly-exact equations.

Time independent methods in semiclassical mechanics: Adiabatic switching

A time independent algorithm based upon Ehrenfest’s adiabatic hypothesis for the construction of the transformation from the physical variables to a set of action-angle variables is presented. At the heart of this algorithm is the recognition that the switching parameter need not be treated as a function of time. The calculations of semiclassical energy levels and dipole matrix elements for the simple quartic oscillator are presented to illustrate this approach.

Time-resolved fluorescence polarization from ordered biological assemblies

We calculate the time dependence of the polarized fluorescence signal from fluorescent-labeled elements of a biological assembly that are rotationally diffusing in an arbitrary three-dimensional angular potential. We have formulated this calculation using the model-independent description of the angular potential wherein the angular potential is described by an expansion in a complete set of orthonormal functions with the expansion coefficients (or order parameters) determined by time-independent methods (Burghardt, T. P., 1984, Biopolymers, 23:2382-2406). We have applied the calculation to fluorescent-labeled myosin cross-bridges in relaxed muscle fibers. In a related paper we describe the experimental observation of the myosin cross-bridges rotationally diffusing in an angular potential (Burghardt, T. P., and N. L. Thompson, 1985, Biochemistry, 24:3731-3735).

Extension of time-independent wave-operator methods to the calculation of the two-body Coulomb photoionization amplitude.

Extension of time-independent wave-operator methods to the calculation of the two-body Coulomb photoionization amplitude.

A general iterative time-independent perturbation theory

An iterative method for time-independent perturbation theory is presented. Lennard-Jones–Brillouin–Wigner (LBW) and Rayleigh–Schrödinger (RS) power series are shown to be particular cases of the iteration and the combined expansion–iteration. Improvements in convergence of the power series are suggested and analyzed.The iterative method gives considerable insight into the nature and relative convergence of the currently used time-independent perturbation methods.

Continuity of wave and scattering operators with respect to interactions

Using time-dependent as well as time-independent methods we establish strong continuity of wave and scattering operators within the two Hilbert space formulation of scattering theory. Our applications to potential scattering include continuity of nonforward total cross sections, scattering amplitudes, and operators for Coulomb-type interactions in R3 and continuity of wave operators for strongly singular potentials in Rn, n≥3.

A global formalism incorporating constants of motion and other constraints in the classical description of chemical rate processes

A phase space cell method for providing a global description of the motions of a physical system is outlined. The key feature is a coarse-graining approximation to the phase-space probability distribution. Constants of motion and other constraints are incorporated to reduce the dimensionality of computations. Time-independent and time-dependent methods of solution are discussed.

A new proof of the invariance principle in scattering theory

For free and interacting Hamiltonians, H 0 and H = H 0 + V( r) acting in L 2(R 3, dx) with V( r) a radial potential satisfying certain technical conditions, and for ϕ a real function on R with ϕ′ > 0 except on a discrete set, we prove that the Moller wave operators Ω± = strong limit e itϕ(H) e −itϕ(H 0) exist and are independent of ϕ. The scattering operator S = (Ω +)∗Ω − is shown to be unitary. Our proof utilizes time independent methods (eigenfunction expansions) and is effective in cases not previously analyzed, e.g. V(r) = sin r r and many others.

Scattering Theory in Quantum Mechanics: Physical Principles and Mathematical Methods

Werner O Amrein, Josef M Jauch and Kalyan Sinha Reading, Massachusetts: W A Benjamin 1911 pp xxiii + 691 price $17.50 (paperback) The late Professor J M Jauch and his coworkers in Geneva have published an authoritative account of the mathematical foundations of quantum mechanical scattering theory in terms of functional analysis in Hilbert space. Following an extensive mathematical introduction, scattering theory is first approached from a timedependent viewpoint before moving to the time-independent methods that are most useful in practical application.

A cognitive multivariate time series and its analysis

The method of analyzing a multivariate time series, by factoring a matrix polynomial of linear transfer function coefficients, is briefly outlined and shown to be potentially applicable to cognitive experiments where the inputs are numerically predictable and the outputs suitably quantified. An experiment on sequences of 90 figures, judged in terms of similarity and of the probability of exceeding or not exceeding some size, was analyzed individually for 12 subjects and their dynamic characteristics expressed in pole zero diagrams. Some dynamic components neglected by classical time-independent psychological methods are identified.

Scattering in an intense radiation field: Time-independent methods

The standard time-independent formulation of nonrelativistic scattering theory is here extended to take into account the presence of an intense external radiation field. In the case of scattering by a static potential the extension is accomplished by the introduction of asymptotic states and intermediate-state propagators which account for the absorption and induced emission of photons by the projectile as it propagates through the field. Self-energy contributions to the propagator are included by a systematic summation of forward-scattering terms. The self-energy analysis is summarized in the form of a modified perturbation expansion of the type introduced by Watson some time ago in the context of nuclear-scattering theory. This expansion, which has a simple continued-fraction structure in the case of a single-mode field, provides a generally applicable successive approximation procedure for the propagator and the asymptotic states. The problem of scattering by a composite target is formulated using the effective-potential method. The modified perturbation expansion which accounts for self-energy effects is applicable here as well. A discussion of a coupled two-state model is included to summarize and clarify the calculational procedures.

Perturbation theory of liquid helium-4 at zero temperature

The formal pertubation theory of a Bose liquid, due to Beliaev, Hugenholtz, Pines, Gavoret, and Nozieres, is carefully examined in an effort to surmount the limitations of the Brueckner-Sawada theory of liquid He 4. In order to make the formal results as directly relevant as possible for Brueckner-theoretic calculations, all of the basic results are rederived by time-independent and number-conserving methods, starting from a linked-cluster expansion for bosons which is the exact analog of the Goldstone expansion for normal fermion systems. The analogy between the ground state and collective excitations of liquid He 4, and those of a normal Fermi liquid, is emphasized and explored in detail. Several aspects of the difficult problem of convergence are discussed. A specific partial summation procedure is recommended, the compact-cluster scheme, to deal with the short-range correlations within clusters of more than two particles. It is then argued that the very strong depletion of the condensate would lead to an extremely slow convergence for the most obvious class of approximations, and a remedy for this difficulty is proposed. The possibility of establishing a fairly detailed correspondence between the perturbation-theoretic wavefunction and the Jastrow wavefunction, including Feenberg-CBF extensions of the latter, is also pointed out.

Quantum Mechanics of Beats between Weakly Coupled Oscillators

Although the weakly coupled double pendulum seems to be a standard illustration in classical mechanics, it is seldom mentioned in quantum mechanics. It shares this fate with the damped harmonic oscillator, but while there is a good reason for avoiding the last one in quantum mechanics, the first can be treated with the same procedure as in classical mechanics. The exact solution is then compared with the solutions obtained by time-independent and time-dependent perturbation methods. It turns out that there are some additional steps to be taken, compared to the usual textbook treatment of the time-independent perturbation theory, due to the fact that all the levels of the unperturbed problem are degenerate. It is shown that the diagonalization of the secular matrix in the time-independent problem is equivalent to a problem of rotation of angular momentum operators in function space. The time-dependent problem needs, again, additional steps due to the degeneracy, and it yields the well-known beat-behavior.

Modified Plane Waves and Rearrangement Collisions

The use of modified plane waves as initial- or final-state basis vectors in the calculation of transition amplitudes for scattering problems is justified in a general manner by time-dependent and time-independent methods. Expressions for the $S$ matrix in terms of modified plane waves are derived for rearrangement collisions. An equivalence theorem is obtained, by means of which the $S$ matrix can be expressed in a variety of alternative forms.

Derivation of the Feynman-Dyson Rules from Time-Independent Theory

The Feynman-Dyson rules for computing the scattering matrix are derived starting with the time-independent Schr\odinger equation. Although no essentially new results are obtained, it is interesting to trace the connection between the time-dependent and time-independent methods starting from the time-independent end rather than, as is usually done, starting from the time-dependent interaction representation. In particular, a theorem due to R. J. Eden plays an important role in our discussion.