Formal scattering theory

We have considered the measurement of observables in electron—atom collisions and the description of the structure of the target and residual atomic states. We are now in a position to develop the formal theory of the reaction mechanism. Our understanding of potential scattering serves as a useful example of the concepts involved. 

Reactions are understood in terms of channels. A channel is a quantum state of the projectile—target system when the projectile and target are so far apart that they do not interact. It is specified by the incident energy and spin projection of the projectile and the quantum state of the A-electron target, which may be bound or ionised. 

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Alhassid, Y., and Levine, R. D. (1985), Formal Scattering Theory by an Algebraic Approach, Phys. Rev. Lett. 54, 739. 

For fixed total energy E, Equation (2.59) defines one possible set of Nopen degenerate solutions I/(.R, r E, n),n = 0,1,2,..., nmax of the full Schrodinger equation. As proven in formal scattering theory they are orthogonal and complete, i.e., they fulfil relations similar to (2.54) and (2.55). Therefore, the (R,r E,n) form an orthogonal basis in the continuum part of the Hilbert space of the nuclear Hamiltonian H(R, r) and any continuum wavefunction can be expanded in terms of them. Since each wavefunction (R, r E, n) describes dissociation into a specific product channel, we call them partial dissociation wavefunctions. 

Unbound systems, such as an electron scattered by a hydrogen atom, are not normalisable, since there is a finite probability of finding the electron anywhere in space. The normalisation of the states of an unbound system will be discussed in chapter 6 on formal scattering theory. 

Methods for calculating collisions of an electron with an atom consist in expressing the many-electron amplitudes in terms of the states of a single electron in a fixed potential. In this chapter we summarise the solutions of the problem of an electron in different local, central potentials. We are interested in bound states and in unbound or scattering states. The one-electron scattering problem will serve as a model for formal scattering theory and for some of the methods used in many-body scattering problems. 

This problem is important in formal scattering theory. A cubic box of side L is represented by a potential F(r), which has a constant value (say zero) inside the box and is infinite at the box boundary. This means that the particle cannot be found outside the box. The coordinate representation of the Schrodinger equation is... 

A formal derivation of (4.134) is given in chapter 6 on formal scattering theory. 

The removal of the ambiguity in the context of formal scattering theory was first achieved by Stelbovics and Bransden (1989) in the case of a one-electron target. They give references to related considerations in a coordinate-space formulation of the problem. The method was extended to a square-integrable representation of the target (e.g. equn. (5.53)) by... 

Use of the potential (7.35) in solving the coupled Lippmann—Schwinger equations (6.73,6.87) corresponding to (7.24) is a unique and numerically-valid description of the electron—atom scattering problem in the context of formal scattering theory. 

The formal scattering theory for describing cosipound-state resonances such as the vibrationally predissociaCing states of interest here, is well established (see, e.g., (32-33) and references therein). For an isolated narrow resonance associated with closed channel m, the S-matrix element between (open) channels j and j is given by (33)... 

Recently, a number of different formulations of resonance Raman scattering have been reported, based on or related to formal scattering theory (Toyozawa, 1976 Hong, 1977). For the present purpose, these treatments do not seem to offer any practical advantages over the methods used here, which are closer to ordinary spectroscopic usage. The formally more sophisticated approaches may be useful when the time dependence of the scattering is being probed. 

Here the second of these expressions is obtained by introducing the transition operator (51). Equation (59) is the Lippmann-Schwinger equation of formal scattering theory, which describe how each monochromatic component of the incoming wave packet is distorted by the scattering interaction (Levine, 1969). 

We outline the formal scattering theory required to define a direct calculation of the IRP. In addition, the ABC modifications to the formal theory are discussed in this Section. 

Because the magnitude of is finite in the ABC formulation, this integral converges in finite time (as opposed to the infinite time required by formal scattering theory). Second, we use the Newton expansion to represent the exponential in Eq. (4.46). Since u(Z) = is an entire analytic function [i.e. u(Z) has no poles], we can... 

A more detailed discussion of this reasoning from formal scattering theory can be found in Ref. 17. One can express the inequality (45) as an equation (46) ... 

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