Reducing a class of two-dimensional integrals to one-dimension with application to Gaussian Transforms

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers times an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions

Analytically Continued Hypergeometric Expression of the Incomplete Beta Function

The Incomplete Beta Function is rewritten as a Hypergeometric Function that is the analytic continuation of the conventional form, a generalization of the finite series, which simpifies the Stieltjes transform of powers of a monomial divided by powers of a binomial

Reduced-mass Fock-Tani Representations for a+ + (b+c-) --\u3e (a+c-) + b+ and First-order Results for {abc} = {ppe, epe, μpμ, μdμ, and μtμ}

The Fock-Tani transformation in the Jacobi three ⟶ two-body reduced-mass system is carried out and the first-order T matrix is found to be identical to that for the full three-body transformation. The Fock-Tani transformation in the reduced-mass system in which particle b is fixed at the origin is found to give a first-order T matrix with an error of mc /mb in the initial momentum wave function. First-order differential and total cross sections are calculated for a+ + (b+c-)⟶(a+c-) + b+ where |abc|= {ppe, epe, μpμ, μdμ, and μtμ}

Reduced Form for the General-state Multicenter Integral from an Integro-differential Transform

In a previous paper Gaussian transforms were utilized to obtain the analytically reduced form for the class of multicenter integrals containing a product of hydrogenic orbitals for s states, Yukawa or Coulomb potentials, and plane waves. In the present paper a related transformation is developed for nonspherical functions, leading to the reduced form for multicenter integrals that include hydrogenic orbitals representing states of arbitrary angular momentum

Analytically Reduced Form Of Multicenter Integrals From Gaussian Transforms

In a previous paper the analytically reduced form was found for the general class of integrals containing multicenter products of ls hydrogenic orbitals, Coulomb or Yukawa potentials, and plane waves. The method consisted of combining all angular dependence within a single quadratic form by means of a three-dimensional Fourier transform and a one-dimensional Feynman transform for each term in the product and an additional integral transformation to move the resulting denominator into an exponential to be summed with the vector products in the plane waves. This quadratic form was then diagonalized with respect to the (introduced) momentum integrals and diagonalized again with respect to the (original) spatial integrals. In the present paper the four-dimensional Fourier-Feynman transformations are replaced by the one-dimensional Gaussian transformation so that only one diagonalization is required, yielding a simpler reduced form for the integral. The present work also extends the result to include all s states and pairs of states with I f\u27=O summed over the m quantum number

Fourier Transform of the Multicenter Product of 1s Hydrogenic Orbitals and Coulomb or Yukawa Potentials and the Analytically Reduced Form for Subsequent Integrals that Include Plane Waves

The Fourier transform of the multicenter product of N 1s hydrogenic orbitals and M Coulomb or Yukawa potentials is given as an (M+N-1)-dimensional Feynman integral with external momenta and shifted coordinates. This is accomplished through the introduction of an integral transformation, in addition to the standard Feynman transformation for the denominators of the momentum representation of the terms in the product, which moves the resulting denominator into an exponential. This allows the angular dependence of the denominator to be combined with the angular dependence in the plane waves

An Integral Transform for quantum amplitudes

The central impediment to reducing multidimensional integrals of transition amplitudes to analytic form, or at least to a fewer number of integral dimensions, is the presence of magnitudes of coordinate vector differences (square roots of polynomials) |x1−x2|2=x21−2x1x2cosθ+x2 √ in disjoint products of functions. Fourier transforms circumvent this by introducing a three-dimensional momentum integral for each of those products, followed in many cases by another set of integral transforms to move all of the resulting denominators into a single quadratic form in one denominator whose square my be completed. Gaussian transforms introduce a one-dimensional integral for each such product while squaring the square roots of coordinate vector differences and moving them into an exponential. Addition theorems may also be used for this purpose, and sometimes direct integration is even possible. Each method has its strengths and weaknesses. An alternative integral transform to Fourier transforms and Gaussian transforms is derived herein and utilized. A number of consequent integrals of Macdonald functions, hypergeometric functions, and Meijer G-functions with complicated arguments is given

First-Order Amplitude for General State-to-State Transitions in Hydrogen by Projectile Impact

The closed analytic form for bound-state transitions due to projectile impact is found in the intermediate representation. The coordinate integral is obtained by evaluating the remaining two integrals in the general multicenter integral derived previously [J.C. Straton, Phys. Rev. A 41, 71 (1990)]. Evaluating the remaining time integral depends upon relating a sum of modified Bessel functions of the second kind KN+1/2(z) to a simple polynomial in 1/z. The results of Van Den Bos and De Heer [Physica 34, 333 (1967)] are shown to be missing a phase factor of (-i)(l′+l

On the Production of the Positive Antihydrogen Ion H̄+ via Radiative Attachment

We provide an estimate of the cross section for the radiative attachment of a second positron into the state of the ion using Ohmura and Ohmura\u27s (1960 Phys. Rev. 118 154) effective range theory and the principle of detailed balance. The ion can potentially be created using interactions of positrons with trapped antihydrogen, and our analysis includes a discussion in which estimates of production rates are given. Motivations to produce include its potential use as an intermediary to cool antihydrogen to ultra-cold (sub-mK) temperatures for a variety of studies, including spectroscopy and probing the gravitational interaction of the anti-atom

A Technique For The Evaluation Of Double Excitation Of Atoms By Fast Protons And Antiprotons

A technique for evaluating cross sections for two-electron excitation in collisions of atoms with fast particles of charge ZP is presented. The atomic wave function is approximated by a sum of pair products of one-electron wave functions, with the coefficients chosen by diagonalizing the fully correlated twoelectron Hamiltonian. Thus spatial correlation is included in both the asymptotic and scattering regions by using these configuration-interaction (Cl) wave functions for initial, intermediate, and final states. Use of CI wave function also allows the first-order contributions to be expressed in closed, analytical form. Both the energy-conserving and energy-nonconserving parts of the second-order amplitude are evaluated. The former (a correlated generalization of the independent-electron approximation) is analytical and the latter is a one-dimensional integral. In helium it is found that the double-excitation cross sections are sensitive to the sign of the projectile charge, but that the energy region where this sensitivity is of the same order as for double ionization is 0.1 to 0.5 Me V /amu, whereas the latter has peak charge sensitivity at 1.5 MeV /amu. Comparison is made with some experimental results