Free Particle Propagator Research Articles

Two-particle problem in a nonequilibrium many-particle system

The two-particle problem within a nonequilibrium many-particle system is investigated in the framework of real-time Green's functions. Starting from the nonequilibrium Bethe-Salpeter equation on the Keldysh contour, a Dyson equation is given for two-time two-particle Green's functions. Thereby the well-known Kadanoff-Baym equations are generalized to the case of two-particle functions. The two-time structure of the equations is achieved in an exact way using the semigroup property of the free-particle propagators. The frequently used Shindo approximation is thus avoided. It turns out that results obtained earlier are valid only in limiting cases of a nondegenerate system or a static interaction, respectively. For the case of thermodynamic equilibrium, the differences to former results obtained for the effective two-particle Hamiltonian are discussed.

Multiple scattering and the Rehr-Albers-Fritzsche formula for the propagator matrix

The propagator matrix is one ingredient in exact theories of multiple scattering. It occurs in the addition theorem (or translation formula) for expanding a spherical outgoing multipole, singular at one point, in terms of regular spherical solutions about another point. It also occurs in the two-centre expansion of the free-space Green's function (or free-particle propagator). Many methods have been devised for computing the propagator matrix, but one of the most efficient, numerically, is based on a formula obtained in 1990 by Rehr and Albers and by Fritzsche. A clear derivation of this formula is given. The formula is also simplified, leading to an expansion in inverse powers of kb, where k is the wavenumber and b is the spacing. This leads to consistent approximations, which are asymptotic as .

Contracted distributed approximating functions: Derivation of non-oscillatory free particle and harmonic propagators for Feynman path integration in real time

Contracted continuous distributed approximating functions (CCDAFs) have been developed. In particular, it has been shown that, continuous distributed approximating functions (CDAFs) based on standard orthogonal polynomials can be contracted to functions formed as the product of a weight function and the sinc function or a Bessel function of the first kind. The CCDAFs of Hermite type have been applied to derive new expressions for the coordinate representation of the free particle evolution operator and that of the evolution operator of harmonic oscillator. These new expressions of free particle and harmonic propagators have as compact mathematical form as Makri’s effective free propagator [N. Makri, Chem. Phys. Lett. 159, 489 (1989)] and Gaussian decay identical to that of the CDAF class free and harmonic propagators due to Kouri et al. [D. J. Kouri, W. Zhu, X. Ma, B. M. Pettitt, and D. K. Hoffman, J. Phys. Chem. 96, 9622 (1992)] and Marchioro et al. [T. L. Marchioro II, M. Arnold, D. K. Hoffman, W. Zhu, Y. Huang, and D. J. Kouri, Phys. Rev. E50, 2320 (1994)], respectively. The Gaussian decay of a CCDAF Hermite free propagator has been shown to be the result of including momentum eigenstates in the propagator which have momenta larger than the momentum of the wave packet of largest momentum that still can be well approximated by the CCDAF considered.

Coulomb-gas approach to quantum motion in random magnetic fields.

We study the motion in the plane of a spinless particle subject to a random magnetic field, in terms of Feynman's influence functional. We show that with \ensuremath{\delta}-correlated fields the Gaussian regime is dominated by one-dimensional quantum motion over classical walks. In the case of long-range field correlations the system is described as a two-dimensional Coulomb gas, where the plasma phase is dominated by free-particle propagation and the low-temperature phase by a regime of strong coupling between advanced and retarded paths. The semiclassical regime is also discussed.

Free particle nonenergy eigenfunctions of a transformed momentum operator using classical solution variables

The classical solution is used to provide appropriate variables for the quantum-mechanical free particle time-dependent Schrödinger wave equation, which then becomes separable giving nonenergy eigenfunctions of closed form. Solutions derived in this fashion may fail to satisfy the homogeneous Schrödinger wave equation at one point in space/time; if this should occur, an inhomogeneous term is produced and a corresponding restricted solution reduces to the free particle propagator. In general, it is shown that the nonenergy eigenfunctions may also be interpreted as eigenfunctions of a time-independent transformed momentum operator. The similarity transformation, by means of which is constructed the transformed momentum operator, the transformed position operator, and the transformed Hamiltonian, are explicitly determined. The relationship of the nonenergy eigenfunctions to the energy eigenfunctions is also exhibited.

(j, 0) ⊕ (0, j) COVARIANT SPINORS AND CAUSAL PROPAGATORS BASED ON WEINBERG FORMALISM

A pragmatic approach to constructing a covariant phenomenology of the interactions of composite high-spin hadrons is proposed. Because there are no known wave equations without significant problems, we propose to construct the phenomenology without explicit reference to a wave equation. This is done by constructing the individual pieces of a perturbation theory and then utilizing the perturbation theory as the definition of the phenomenology. The covariant spinors for a particle of spin j are constructed directly from Lorentz invariance and the basic precepts of quantum mechanics following the logic put forth originally by Wigner and developed by Weinberg. Explicit expressions for the spinors are derived for j=1, 3/2 and 2. Field operators are constructed from the spinors and the free-particle propagator is derived from the vacuum expectation value of the time-order product of the field operators. A few simple examples of model interactions are given. This provides all the necessary ingredients to treat at a phenomenological level and in a covariant manner particles of arbitrary spin.

A computational demonstration of the distributed approximating function approach to real time quantum dynamics

The authors report computational applications for the newly developed distributed approximating function (DAF) approach to real time quantal wavepacket propagation for several one-dimensional model problems. The DAF is constructed to fit all wavepackets accurately which can be represented, to the same accuracy, by a polynomial of degree M, or less, within the envelope of the DAF. (This defines the {open_quotes}DAF class{close_quotes} of functions.) By expressing the DAF (and thus the wavepacket to be propagated) in terms of Hermite functions (each a product of a Hermite polynomial and its Gaussian generating function), the DAF approximation to the wavepacket is propagated freely and exactly for a short time {tau}. The Hermite functions are the natural basis states for describing the free evolution of a localized particle and yield a highly banded representation for the free particle propagator. Combining the DAF class free propagation scheme with any of several short time approximations to the full propagator enables one to propagate the wavepacket through a potential. The DAF results for the propagated wavepacket and various scattering amplitudes are shown to be in good agreement with those obtained by more standard methods. 17 refs., 7 figs., 4 tabs.

Convolutions of particle Green’s functions

The use of Feynman causal function in the perturbative treatment of theS-matrix made the computation of convolutions an easy and well-known procedure for free-particle propagators. But the convolution of its components, like theδ and principal values among themselves, is very rarely looked upon. In field theories with higher-order equations of motion some of these convolutions appear as the fundamental ingredients. A discussion of these convolutions is explicitly done in the simplest examples.

Distributed approximating function theory: a general, fully quantal approach to wave propagation

A new procedure for developing a coarse-grained representation of the free particle propagator in Cartesian, cylindrical, and spherical polar coordinates is presented. The approach departs from a standard basis representation of the propagator and the state function to which it is applied. Instead, distributed approximating functions (DAFs), developed recently in the context of propagating wave packets in 1-D on an infinite line, are used to create a coarse-grained, highly banded matrix which produces arbitrarily accurate results for the free propagation of wave packets

Quadrature-based, coarse-grained treatment of the coordinate representation free particle real-time evolution operator

In this paper we report a quadrature evaluation of the coordinate representation, short-time free particle propagator, 〈R‖exp(−iH0τ)‖R′〉. The result is the elimination of most of the highly oscillatory behavior in this quantity yielding in its stead a much smoother function, strongly peaked at R=R′. We view this as a numerical coarse graining of the propagator which leads to the intuitively reasonable result that for short times τ or large mass, the particle should not have a significant amplitude for R points that are far from R′. This leads to an interesting, and potentially useful, banded structure for 〈R‖exp(−iH0τ)‖R′〉. Calculations have been carried out both for zero and nonzero orbital angular momenta, for which we also give the exact analytic results, and the same behavior is found. The quadrature-coarse graining procedure still appears to retain the important quantum effects as demonstrated by subsequent use of the coarse-grained free propagator to calculate the scattering of an electron by a simple central potential. Results are in quantitative agreement with those obtained by alternative, numerically exact methods. The coarse-grained free propagator is, of course, independent of the potential, and we expect that it can provide a very useful tool for computing real-time dynamics for a variety of systems.

An asymptotic wavefunction splitting procedure for propagating spatially extended wavefunctions: application to intense field photodissociation of H +2

We describe an efficient procedure for propagation a spatially extended wavefunction evaluated on a small grid. The procedure involves the repeated splitting of the wavefunction into a part residing mainly in the interaction region (small fragment separation) and parts residing solely in the asymptotic region (large fragment separation), and propagation each part separately. The splitting is done by multiplying the wavefunction by the functions f( R) and 1− f( R), where R is the internuclear separation of an unbound degree of freedom and f(R) equals one in the interaction region and falls to zero in the asymptotic region. Each asymptotic piece of the wavefunction is propagated to some final time, in the momentum representation where it does not spread or translate, by a single application of a free-particle propagator. Unlike the use of an absorbing boundary, the total wavefunction can be reassembled at a later time by adding the individual asymptotic pieces, propagated to a common final time with a free-particle propagator, to the part of the wavefunction remaining in the interaction region. This procedure is well suited to propagation methods that evaluate the wavefunction on a spatial grid, because the amount of asymptotic region included in the grid, and thereforethe computational effort, can be minimized (the smaller the asymptotic region, the more frequent the splitting). The propagation method is demonstrated in the calculation of the fragment kinetic energy distribution produced by the intense field photodissociation of H + 2. This example illustrates the propagation method for a system whose wavefunction becomes very extended with time due to multiphoton bound-continuum and continuum-continuum transitions.

Multiple-scattering Green-function method for electronic-structure calculations of surfaces and coherent interfaces.

We present a formally exact solution of the multiple-scattering-theory (MST) equations for the case of semi-infinite and coherent doubly semi-infinite materials. For the case of semi-infinite materials (surfaces), we also present an alternative approach based upon the embedded-cluster method, which can be numerically as accurate as the exact method but is computationally easier to implement. The methods presented here satisfy the correct boundary conditions associated with semi-infinite materials and thus constitute a proper generalization of the Green-function formalism as applied to bulk systems, to the treatment of surfaces and interfaces. Specifically, the lack of translational invariance in directions perpendicular to the surface or interface is handled through the solution of a self-consistent equation for the scattering matrix, describing exactly the interaction of any chosen plane near the surface or interface with the rest of the material. The formalism is free of the introduction of extraneous conditions such as the artificial truncation of the free-particle propagator or the range of electron hopping elements, or the use of a complex potential. Furthermore, being based on a first-principles, MST Green-function method, the formalism is applicable to the treatment of a large number of physically important problems such as surface and/or interface regions formed by pure materials or concentrated alloys, the study of impurities of diverse kinds near such a region, and many others. Results for one-dimensional model systems as well as realistic three-dimensional materials are presented. The advantages, convergence properties, and restrictions of the formalisms are discussed and further work, currently in progress, is commented upon.

Relativistic second-order Born theory for electron capture.

The asymptotic dependence of the second-order Born term on the collision energy E is reexamined. In contrast to previous work, a relativistic free-particle propagator is used, and allowance is made for the spectator electrons of the target by means of a screened potential. These modifications do not change the behavior of the capture cross section like (lnE${)}^{2}$/E.

An efficient procedure for calculating the evolution of the wave function by fast Fourier transform methods for systems with spatially extended wave function and localized potential

Various methods using fast Fourier transform algorithms or other ‘‘grid’’ methods for solving the time-dependent Schrödinger equation are very efficient if the wave function remains spatially localized throughout its evolution. Here we present and test an extension of these methods which is efficient even if the wave function spreads out, provided that the potential remains localized. The idea is to split the wave function at various times during the propagation into two parts, one localized in the interaction region and the other in the force free region; the first is propagated by a fast Fourier transform method on a grid whose size barely exceeds the interaction region, and the latter by a single application of a free particle propagator. This splitting is performed whenever the interaction region wave function comes close to the end of the grid. The total asymptotic wave function at a given time t is reconstructed by adding coherently all the asymptotic wave function pieces which were split at earlier times, after they have been propagated to the common time t. The method is tested by studying the wave function of a diatomic molecule dissociated by a strong laser field. We compute the rate of energy absorption and dissociation and the momentum distribution of the fragments.

Cosmology with time-varyingG

It is shown that a Universe with a time-varying gravitational “constant”G necessarily implies creation if the rest mass of matter particlesm p is constant. In this case, from Einstein's field equations, the conditions for energy-momentum propagation are ∇ ·(GT μv ) from which matter and photon propagation equations are derived. Free matter particle propagation is not affected by creation that is given byGN pmp=const, whereN p is the number of matter particles within a proper volume. This relation introduces explicitly the rest mass of the Universe into the field equations. Free photon propagation is affected by creation that is given byGT λ v R=const, whereN λ is the number of photons within a proper volume, which is the cosmic red shift law. Conservation of the cosmic background photon distribution determines photon creation asG 3 N λ 4 . The results are applied to the caseG ⩜t −1 equivalent toN p ÷ t. It is found that at an aget=1, 0−40 t o, of the order light takes to travel a proton size, Planck's units become of the order of the proton's massm p, sizer p, and timer p/c. Hence, matter particles at this age are quantum black holes. Evaporation of these quantum black holes at this age gives a background blackbody radiation that, red shifted to present timet 0, gives the present cosmic microwave background. A cosmological model of the Friedmann type is constructed. The red shift versus distance relation is derived taking into account creation. Using a Hubble's constantH obs=50 km sec−1 Mpc−1 and a deceleration parameterq obs=1.0 the model is of the typek=1 and gives a present aget 0=6.81×109 yr, consistent with Uranium model ages. Thus, the three results for the age of the Universe, i.e., radioactive decay, Hubble's constant, and stellar evolution are brought together in this creation model. The matter-dominated era occurs fort>7.6×10−3 t 0, while the radiation-dominated era occurs for 7.6×10−3 t o>t>10−40 t o. The origin of the Universe is placed at this last limit, which is Planck's time at the corresponding G, consisting of quantum black holes at a temperature Ti≃=3×1011K.

Multiple scattering in the Lee model

It is shown that the amplitude for the scattering of a theta particle from an arbitrary number of fixed N particles in the Lee model can be obtained from Watson's multiple scattering theory if the elementary interaction is taken to be the full theta-N T matrix, including the V-particle pole, and the propagator is taken to be a free particle propagator.

Path integration of a generalized quadratic action

Path integration of a class of generalized quadratic actions first proposed by Feynman is performed within the framework of Feynman's polygonal path approach. The exact propagator has the form of a free particle propagator with an “effective mass” apart from the normalization factor. Relation between the propagator and the usual Van Vleck-Pauli formula is discussed.

Evaluation of the second Born amplitude as a two-dimensional integral for H++H(1s) to H(1s)+H+

The second Born amplitude (with the exact propagator replaced by the free particle propagator) for the electron capture reaction H++H(1s) to H(1s)+H+ is reduced to a two-dimensional integral. The integrand has several singularities, which are dealt with. Results of the numerical integration of the cross section for various energies in the 25-200 keV range are presented.

Total inelastic-scattering cross sections by optical-potential methods

Total excitation cross sections for electrons scattering from atomic hydrogen are calculated by using a complex optical potential. Distorted-wave propagators as well as free-particle propagators are used in computing the optical potential. It is found that using an attractive potential to generate the propagator results in poorer agreement with experiment than using a free-particle propagator. The effect of neglecting coupling between excited target states in computing the optical potential is also studied. It is concluded that it is necessary to include coupling between excited target states if the calculation is to be reliable.

Covariant WKB expansion in Riemannian space

We show how to compute in a covariant was explicitly the higher order corrections in the semiclassical expansion of the free particle propagator in curved spaces.