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Register a new userPartial-wave Coulomb t-matrices for like-charged particles at ground-state energy
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Added by
Vladislav Kharchenko
Publication date
2017
fields
Physics
and research's language is
English
Authors
V. F. Kharchenko
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We study a special case at which the analytical solution of the Lippmann-Schwinger integral equation for the partial wave two-body Coulomb transition matrix for likely charged particles at negative energy is possible. With the use of the Focks method of the stereographic projection of the momentum space onto the four-dimensional unit sphere, the analytical expressions for s-, p- and d-wave partial Coulomb transition matrices for repulsively interacting particles at bound-state energy have been derived.
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Starting from the integral representation of the three-dimensional Coulomb transition matrix elaborated by us formerly with the use of specific symmetry of the interaction in a four-dimensional Euclidean space introduced by Fock, the possibility of the analytical solving of the integral equation for the partial wave transition matrices at the excited bound state energy has been studied. New analytical expressions for the partial s-, p- and d-wave Coulomb t-matrices for like-charged particles and the expression for the partial d-wave t-matrix for unlike-charged particles at the energy of the first excited bound state have been derived.
Leaning upon the Fock method of the stereographic projection of the three-dimensional momentum space onto the four-dimensional unit sphere the possibility of the analytical solving of the Lippmann-Schwinger integral equation for the partial wave two-body Coulomb transition matrix at the ground bound state energy has been studied. In this case new expressions for the partial p-, d- and f-wave two-body Coulomb transition matrices have been obtained in the simple analytical form. The developed approach can also be extended to determine analytically the partial wave Coulomb transition matrices at the energies of excited bound states. Keywords: Partial wave Coulomb transition matrix; Lippmann-Schwinger equation; Fock method; Analytical solution PACS Nos. 03.65.-w; 03.65.Nk; 34.20.Cf
Several ultra-compact accurate wave functions in the form of generalized Hylleraas-Kinoshita functions and Guevara-Harris-Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), or, equivalently, the Non-Relativistic QED (NRQED), for the ground state energies (4-5 significant digits (s.d.)) of He-like and Li-like iso-electronic sequences in the static approximation with point-like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in $Z$ by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He-like two-electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy $sim 10^{-3}$,a.u. and the same relative accuracies $sim 10^{-2}-10^{-3}$ for both the cusp parameters and the six expectation values, is found. For the Li-like three-electron sequence the most accurate ultra-compact function taken as the variational trial function provides absolute accuracy for energy $sim 10^{-3}$,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for $Z leq 20$ and 3 s.d. for the two expectation values for $Z=3$.
Low-energy elastic and inelastic scattering in the Ps(1$s$)-Ps(2$s$) channel is treated in a four-body hyperspherical coordinate calculation. Adiabatic potentials are calculated for triplet-triplet, singlet-singlet, and singlet-triplet spin symmetries in the spin representation of coupled electrons and coupled positrons, with total angular momentum $L=0$ and parity equal to $+1$. The s-wave scattering lengths for the asymptotic Ps(1$s$)-Ps(2$s$) channel are calculated for each spin configuration. Results obtained for the s-wave scattering lengths are $a_{mathrm{TT}}=$~$7.3(2)a_0-i0.02(1)a_0$, $a_{mathrm{SS}}=$~$13.2(2)a_0-i0.9(2)a_0$, and $a_{mathrm{ST}}=$~$9.7(2)a_0$ for each spin configuration. Spin recoupling is implemented to extract the scattering lengths for collisions of Ps in different spin configurations through properly symmetrized unitary transformations. Calculations of experimentally relevant scattering lengths and cross-sections are carried-out for Ps atoms initially prepared in different uncoupled spin states.
Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized hypergeometric function $_2F_2(a,a;a+l+1,a-l;z)$ is expressed with the help of confluent hypergeometric functions and the asymptotic expansion of $_2F_2(a,a;a+l+1,a-l;z)$ is simplified.
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