Integral equations momentum space

One Important aspect of the supercomputer revolution that must be emphasized Is the hope that not only will It allow bigger calculations by existing methods, but also that It will actually stimulate the development of new approaches. A recent example of work along these lines Involves the solution of the Hartree-Fock equations by numerical Integration In momentum space rather than by expansion In a basis set In coordinate space (2.). Such calculations require too many fioatlng point operations and too much memory to be performed In a reasonable way on minicomputers, but once they are begun on supercomputers they open up several new lines of thinking. 

The equations to be fulfilled by momentum space orbitals contain convolution integrals which give rise to momentum orbitals ( )(p-q) shifted in momentum space. The so-called form factor F and the interaction terms Wij defined in terms of current momentum coordinates are the momentum space counterparts of the core potentials and Coulomb and/or exchange operators in position space. The nuclear field potential transfers a momentum to electron i, while the interelectronic interaction produces a momentum transfer between each pair of electrons in turn. Nevertheless, the total momentum of the whole molecule remains invariant thanks to the contribution of the nuclear momenta [7]. 

The function 4> k) is known as the wave function in momentum space. The Fourier integral represents the superposition of many waves of different wave vectors. This construct defines a wave packet, once considered as the theoretically most acceptable description of a wave-mechanical particle5. Schrodinger s dynamical equation (4) for a free particle... 

Equation (24) can be derived from the theory of hyperspherical harmonics and Gegenbauer polynomials but for readers unfamiliar with this theory, the expansion can be made plausible by substitution into the right-hand side of equation (23). With the help of the momentum-space orthonormality relations, (17), it can then be seen that right-hand side of (23) reduces to the left-hand side, which must be the case if the integral equation is to be satisfied. Let us now consider an electron moving in the attractive Coulomb potential of a collection of nuclei ... 

The counterpart wavefunction in momentum-space, 4>(yi,y2 is a function of momentum-spin coordinates % = (jpk, k) in which pk is the linear momentum of the feth electron. There are three approaches to obtaining the momentum-space wavefunction, two direct and one indirect. The wavefunction can be obtained directly by solving either a differential or an integral equation in momentum- or p space. It can also be obtained indirectly by transformation of the position-space wavefunction. 

It is known that the Schrodinger equation in momentum space takes the form of an integral equation ... 

The formulation outlined above is in configuration space, but several authors, notably Ghosh and his collaborators (Chaudhury, Ghosh and Sil, 1974) and Mitroy (1993), also Mitroy, Berge and Stelbovics (1994) and Mitroy and Ratnavelu (1995), have preferred to work in momentum space with a set of coupled integral equations rather than the coupled integro-differential equations (3.31) and (3.32). 

We first consider the sum of states. Now, in Eq. (A.33) the integration over coordinates gives the volume of the container, and the integral over the momenta is the momentum-space volume for H having values between 0 and E. Equation (A.41) is the equation for a sphere in momentum space with radius j2rn, 11. Thus, the volume of the sphere is 4Tt(y/2mH)3/3 and... 

A large breakthrough in physical transparency and ease of computation is achieved by expressing the problem as an integral equation in momentum space. The reason for this is that scattering experiments measure momenta, not positions, so that the momentum-space description parallels the experiment. 

The momentum-space integral equation is the Lippmann—Schwinger equation. It is an equation for the T matrix. We multiply (4.I0I) on the left by V, take the matrix element for the eigenstate (k of the final momentum, and introduce the spectral representation of K. 

The Lippmann—Schwinger equations (6.73) are written formally in terms of a discrete notation i) for the complete set of target states, which includes the ionisation continuum. For a numerical solution it is necessary to have a finite set of coupled integral equations. We formulate the coupled-channels-optical equations that describe reactions in a channel subspace, called P space. This is projected from the chaimel space by an operator P that includes only a finite set of target states. The entrance channel 0ko) is included in P space. The method was first discussed by Feshbach (1962). Its application to the momentum-space formulation of electron—atom scattering was introduced by McCarthy and Stelbovics... 

Using the Dirac delta function to perform the first / -integration in (82), we obtain the momentum-space Schrodinger equation ... 

There are three distinct ways by which the momentum-space wave function can be obtained directly by solving either a differential or an integral equation in momentum or p space, or indirectly by transformation of the position-space wave function. 

Equation (31) shows a characteristic feature of the resonance wavefunc-tions, namely that the integral over radial space (not over the spherical harmonics), involves the square of the function itself and not of its absolute value. However, due to the asymptotic behavior (25a), the integral (31) (as well as the one of Eq. (30)), is infinite, since the integrant goes like where a is the imaginary part of the complex momentum of the free Gamow orbital. 

As we have already mentioned above, for 5-wave scattering the function/(r,R) depends only on the absolute values of r and R and on the angle between them. Thus, Equation 10.24 is an integral equation for the function of three variables. In order to find the molecule-molecule scattering length, it is more convenient to transform Equation 10.24 into an equation for the momentum-space function, /(k,p) = / d rd R/(r,R)exp(ik r/a-l-ip R/ s/2a), which yields the following expression ... 

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