Integral equation for the box-normalised collision state

Note that in the time-reversed situation the beam is switched off over a finite period x and the weight factor of the wave packet is e.  

The experimental situation is that times characteristic of an experiment are of the order 10 s, while the time characteristic of an atomic collision is, for example, the time it takes an electron projectile to traverse an atom. This is of the order 10 s. It is therefore physically reasonable to consider the limit t — oo or e — 0+. Experiments involving time resolution have been devised with resonant states whose lifetimes are greater than 10 s. Such experiments must be described by explicit wave packets rather than the formalism of the present section. An example is given in section 4.6. 

The double limit e — 0+, L — oo must be taken in such a way that the whole system is inside the normalising box at all times, x is the length of the incident train of particles divided by their velocity v. We require 

We have used (6.5) to introduce the stationary channel state. From 

We use (6.22) to obtain from (6.21) the box-normalised wave-packet collision state at t = 0. 

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