Exact solution of coupled equations and the hyperspherical formalism: Calculation of expectation values and wavefunctions of three Coulomb-bound particles

Exact solutions of one-dimensional coupled differential equations are developed by substituting in power series. The properties of these solutions and the possibility of their application to the few-body problem in the framework of the hyperspherical method are studied. The necessity of logarithmic terms in the nonrelativistic many-body wavefunctions, as well as their absence in the relativistic case, is stressed. Explicit form of the solution of the one-dimensional hyperspherical matrix equation corresponding to the three-body Coulomb problem is found and used to obtain Schroedinger and Faddeev bound state wavefunctions, correlation integrals and probabilities of different hyperspherical states. The results of calculations with inclusion of up to 25 hyperspherical harmonics (K_m = 16) for the ground and excited state of the helium atom, the ground state of the positronium ion and the negative hydrogen ion are given and compared with those obtained by the multiconfigurational Hartree-Fock and variational methods as well as with other hyperspherical calculations. We find that generally the correlation integrals converge as the energies, that is, as $\text{1}\text{K}{}_{\mathrm{<mtext>m</mtext>}}{}^4$. While the method is essentially exact, computer round-off error limits the precision for K_m > 12 in the positronium calculations.     

Beyond the first order of the Hyperspherical Harmonic Expansion Method

In this paper we demonstrate the inadequacy of the first order of the Hyperspherical Harmonic Expansion Method, the L_m approximation, for the calculation of the binding energies, charge form factors and charge densities of doubly magic nuclei like ¹⁶O and ⁴⁰Ca. We then extend the Hyperspherical Expansion Method to many-fermion systems, consisting of an arbitrary number of fermions, and develop an exact formalism capable of generating the complete optimal subset of the hyperspherical harmonic basis functions. This optimal subset consists of those hyperspherical harmonic basis functions directly connected to the dominant first term in the expansion, the hyperspherical harmonic of minimal order L_m, through the total interaction between the particles. The required many-body coefficients are given using either the Gogny or Talmi-Moshinsky coefficients for the two-body operators. Using the two-body coefficients the weight function generating the orthogonal polynomials associated with the optimal subset is constructed.     

Ro-vibrational states of triplet H3 (aΣu): The lowest 19 bands

We have performed extensive calculations of the ro-vibrational states of triplet H₃⁺, using the method of hyperspherical harmonics and our recently reported double many-body expansion potential energy surface. The rotational term values of the lowest 19 states are presented here for a total angular momentum of J?10.     

A set of hyperspherical harmonics especially suited for three-body collisions in a plane

We obtain a set of four-dimensional hyperspherical harmonics in closed form. These harmonics are not only quantized with respect to the rotation group (O ₂), but are an irreducible basis for the permutation groupS ₃. An additional symmetry is found which allows us to write hyperspherical harmonics classified with respect to a 12 element groupS ₃×i×O ₂. We give a set of three mutually commuting operators whose eigenvalues uniquely characterize each spherical harmonic with respect to degree, symmetry, and angular momentum in the plane.     

Low-Lying S-States of Two-Electron Systems

The energies of the low-lying bound S-states of some two-electron systems (treating them as three-body systems) like negatively charged hydrogen, neutral helium, positively charged-lithium, beryllium, carbon, oxygen, neon, argon and negatively charged muonium and exotic positronium ions have been calculated employing hyperspherical harmonics expansion method. The matrix elements of two-body interactions involve Raynal–Revai coefficients which are particularly essential for the numerical solution of three-body Schr?dinger equation when the two-body potentials are other from Coulomb or harmonic. The technique has been applied for to two-electron ions ¹H^? (Z = 1) to ⁴⁰Ar¹⁶⁺ (Z = 18), negatively charged-muonium Mu^? and exotic positronium ion Ps^?(e ⁺ e ^? e ^?) considering purely Coulomb interaction. The available computer facility restricted reliable calculations up to 28 partial waves (i.e. K _m  = 28) and energies for higher K _m have been obtained by applying an extrapolation scheme suggested by Schneider.     

Elliptic PDE formulation and boundary conditions of the spherical harmonics method of arbitrary order for general three-dimensional geometries

The inherent complexity of the radiative transfer equation makes the exact treatment of radiative heat transfer impossible even for idealized situations and simple boundary conditions. Therefore, a wide variety of efficient solution methods have been developed for the RTE. Among these solution methods the spherical harmonics method, the moment method, and the discrete ordinates method provide means to obtain higher-order approximate solutions to the equation of radiative transfer. Although the assembly of the governing equations for the spherical harmonics method requires tedious algebra, their final form promises great accuracy for any given order, since it is a spectral method (rather than finite difference/finite volume in the case of discrete ordinates). In this study, a new methodology outlined in a previous paper on the spherical harmonics method (P_N) is further developed. The new methodology employs successive elimination of spherical harmonic tensors, thus reducing the number of first-order partial differential equations needed to be solved simultaneously by previous P_N approximations (=(N+1)²). The result is a relatively small set (=N(N+1)/2) of second-order, elliptic partial differential equations, which can be solved with standard PDE solution packages. General boundary conditions and supplementary conditions using rotation of spherical harmonics in terms of local coordinates are formulated for the general P_N approximation for arbitrary three-dimensional geometries. Accuracy of the P_N approximation can be further improved by applying the “modified differential approximation” approach first developed for the P₁-approximation. Numerical computations are carried out with the P₃ approximation for several new two-dimensional problems with emitting, absorbing, and scattering media. Results are compared to Monte Carlo solutions and discrete ordinates simulations and a discussion of ray effects and false scattering is provided.     

An essentially exact many body calculation for muonic molecular ions and exotic coulombic systems

The L = 0 bound states and the structural properties of muonic molecular ions and exotic nuclear molecular ions are calculated by a hyperspherical harmonics expansion method, which is an essentially exact method for three-body systems. We examine the convergence rate of this method and also examine its crucial dependence on the mass ratio ? (ratio of the masses of similar to dissimilar particles). An a priori criterion for the rate of convergence and the required size of the expansion basis for a predetermined precision have been suggested. Our calculation reveals the cluster type of structure of these systems.     

Structure of A = 3 nuclear systems using realistic Hamiltonians

The structure of A = 3 low-energy scattering states is described using the hyperspherical harmonics method with realistic Hamiltonian models, consisting of two- and three-nucleon interactions. Both coordinate and momentum space two-nucleon potential models are considered.     

From three nucleons to three quarks

Some short history of three-body methods originated from the famous Skornyakov-Ter-Martirosyan equation is given, including the latest development of Faddeev formalism and Efimov states. The 3q system is shown to require an alternative, which is provided by the hyperspherical method (K harmonics), highly successful for baryons.     

Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous,anisotropically scattering media

Radiative transfer in a three-dimensional rectangular enclosure containing radiatively participating gases and particles is studied using the first- and third-order spherical harmonics approximations. Inhomogeneities in the radiative properties of the medium, as well as in the radiation characteristics of the boundaries, are allowed for. The scattering phase function is represented by the delta-Eddington approximation, and it is assumed to be a function of the location in order to account for density variation of the particles in the medium. Numerical solutions of the model equations are obtained using a finite difference scheme. For the purpose of validating the P₃-approximation, the results are compared with those based on Hottel's zonal method.     

Theoretical Study of the 4He(γ, p)3H and 4He(γ, n)3He Reactions

Ab initio calculation of the total cross section for the reactions ⁴He(γ, p)³H and ⁴He(γ, n)³He is presented, using state-of-the-art nuclear forces. The Lorentz integral transform (LIT) method is applied, which allows exact treatment of the final state interaction (FSI). The dynamic equations are solved using the effective interaction hyperspherical harmonics method. In this calculation of the cross sections the three-nucleon force is fully taken into account, except in the source term of the LIT equation for the FSI transition matrix element.     

Method for solving the many-body bound state nuclear problem

We present a method based on hyperspherical harmonics to solve the nuclear many-body problem. It is an extension of accurate methods used for studying few-body systems to many bodies and is based on the assumption that nucleons in nuclei interact mainly via pairwise forces. This leads to a two-variable integro-differential equation which is easy to solve. Unlike methods that utilize effective interactions, the present one employs directly nucleon-nucleon potentials and therefore nuclear correlations are included in an unambiguous way. Three body forces can also be included in the formalism. Details on how to obtain the various ingredients entering into the equation for the A-body system are given. Employing our formalism we calculated the binding energies for closed and open shell nuclei with central forces where the bound states are defined by a single hyperspherical harmonic. The results found are in agreement with those obtained by other methods.     

Phonon–particle coupling effects in odd–even double mass differences of semi-magic nuclei

A method is developed to consider the particle–phonon coupling (PC) effects in the calculation of the odd–even double mass differences (DMD) in semi-magic nuclei starting from the free NN potential. The PC correction δΣ^PC to the mass operator Σ is found in g _L ²-approximation, g _L being the vertex of creating the L-phonon. The tadpole term of the operator δΣ^PC is taken into account. The method is based on a direct, without any use of the perturbation theory, solution of the Dyson equation with the mass operator Σ(ε) = Σ₀ + δΣ^PC(ε) for finding the single-particle energies and Z-factors. In its turn, they are used as an input for finding different PC corrections to the DMD values. Results for a chain of even semi-magic nuclei ^200?206Pb show that the inclusion of the PC corrections makes agreement with the experimental data significantly better.     

The hyperspherical harmonics method applied to 12C

We apply the hyperspherical harmonics method to the low-lying T = 0 states in ¹²C, using the 3-α model. With the results from ref. ¹) we are able to investigate all states up to the 4⁺ level at 14.1 MeV, except for the broken symmetry 1⁺ state at 12.7 MeV. Antisymmetrization effects are included by adopting the local potential approximation²). Energy levels and electron scattering form factors are calculated.     

Local realistic NN interactions in the four-nucleon problem

The ⁴He problem with realistic NN interactions is solved and systematically studied. All such interactions are found to underbind ⁴He significantly. Possible sources of this underbinding are discussed. The method of hyperspherical harmonics is used.     

Correlated hyperspherical harmonics

The expansion of the wavefunction for a bound three particle state in the five-dimensional hyperspace of hyperspherical harmonics in some cases suffers bad convergence, especially for weakly bound states. For this reason correlated hyperspherical harmonics are proposed, of which the ordinary hyperspherical harmonics are one special choice. The “best” suited correlated hyperspherical harmonics are chosen from an infinite set of complete orthogonal systems by a Ritz variational calculation.     

Chiral symmetry breaking and condensates in the rishon model

The rishon model is studied in the limit g_c → 0, α → 0 when its global flavour symmetry is SU(6) × SU(6) × U(1) analogous to six massless flavour QCD. Recently it was shown that the ad hoc breaking SU(6) × SU(6) → SU(3) × SU(3) allows the anomaly constraint to be satisfied. In this paper this is shown to be but one of several successful patterns of chiral symmetry breaking. The condensates required to perform these breakings are fully discussed. A plausibility argument based on single gauge boson exchange is presented which determines the condensate uniquely to be 〈(v_LV_L)³〉 corresponding to the original breaking above. The same argument applies to QCD, which is argued to differ in its chiral behaviour due to the large intrinsic masses of the quarks. The implications of the above condensate and pattern of chiral symmetry breaking for the rishon model include the prediction of integer charged colour octet fermions, a naive mass formula m_e = 2m_u ? m_d, new insight into the parity-violating condensate 〈(v_Lv_L)²(v_Rv_R)〉 and the prediction of 52 new pseudos whose masses are estimated.     

A heavy lepton mechanism yielding electrons in pp collisions at high energies

Heavy leptons L₁, L₂ with strong interactions (lumotons) may give rise to large transverse momentum electrons in pp collisions. If the associated neutrino v_L is stable and massive, m(v_L) ? m(L₁), the L₁ decays will yield electrons peaked at small transverse momenta. v_L detection is discussed.     

Elliptic dichroism, ellipticity and the offset angle of high harmonics generated by arbitrary homonuclear diatomic molecules

The nth harmonic emission rate has contributions of the components of the T-matrix element in the direction of the laser-field polarization and in the direction perpendicular to it. Using both components of the T-matrix element we present a theoretical approach for calculation of the ellipticity and the offset angle of high harmonics. The molecular bound state is represented by HOMO or by HOMO-1. We show that high harmonics, generated by molecules oriented by an angle ??_L with respect to the major semiaxis of the laserfield polarization ellipse, are elliptically polarized even if the applied field is linearly polarized. Using examples of N₂ and O₂ molecules we show the existence of extrema and sudden changes of the harmonic ellipticity and the offset angle for particular molecular alignment. The interference between different contributions to the T-matrix element depends on the molecular symmetry. Presenting partial or total parameters of elliptic dichroism in the (??_L, n) plane clear interference minima are observed. Therefore, the measurement of the elliptic dichroism may reveal information about the molecular structure and symmetry.