Scattering theory target states

The time spectrum of the scattered electron in this limit has the shape g-Frf/fi jjjg lifetime of the compound electron—target system in the resonant state is Xr. We have derived the same result for a detailed scattering theory that we knew already from the uncertainty principle. The uncertainty relation (4.146) becomes an exact equality (4.167) if Tr is the full width at half maximum and Xr is the lifetime. 

Scattering theory concerns a collision of two bodies, that may change the state of one or both of the bodies. In our application one body (the projectile) is an electron, whose internal state is specified by its spin-projection quantum number v. The other body (the target) is an atom or an atomic ion, whose internal bound state is specified by the principal quantum number n and quantum numbers j, m and / for the total angular momentum, its projection and the parity respectively. We... 

The scattering problem is formulated in terms of one-electron states, which we call orbitals to distinguish them from the A/ -electron target states and the (AZ -l-l)-electron collision and channel states of scattering theory. The space of collision states is spanned by products of N+l orbitals, which we explicitly antisymmetrise in this section. 

The structure of MQDT involves a compact target state within a radius ro and an asymptotic solution, valid at large r, which is joined to the wavefunction of the inner region in order to determine amplitudes and phases. Thus, MQDT is simply a special form of scattering theory, specially adapted to handle bound Rydberg states of atoms and adjacent structure in the base of the ionisation continuum. 

Equations (61) and (65) describe the time-dependent response of the molecule to the incoming photon states. In practice, this response is most conveniently studied by detecting the scattered light component (65) far away from the target. Formally, we may then follow the standard prescription in scattering theory and use the asymptotic form of the function i/ r (0>, taken in the limit t = -i- oo. From the second of Eqs. (65), we then get (Shore, 1967)... 

State where one electron is bound and the other free. Worse yet, the channel states in scattering theory that need to be constructed can only be identified by an asymptotic boundary condition, a region where a finite Hilbert basis representation will be inaccurate. What is needed is a method which relies on information about the wave function in the target region to accurately identify and separate processes which are only truly separable in the asymptotic region. This is the interpretation problem. 

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. 

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