Box-normalised wave-packet states

We are interested in the rate of transition from the entrance channel o(t)) to the exit channel ,(0) caused by the interaction V. We choose t = 0 to be the time at which the collision occurs and consider the mathematical representation of the dependence of the collision state Fo(t)) on o(T)), its state at time T in the distant past (on an atomic scale) when the system was so well separated that the potential V did not act. Applying the time-development operator (3.45) we have 

However, this does not represent the physical situation. The system does not start suddenly at T. The beam is switched on over a finite period t. We therefore represent Fo(t) as the time development of a wave packet prepared with an exponentially-increasing weight e, where 

The exponential form is chosen for mathematical convenience, since it leads to an integral that can be evaluated. The exact form of the wave 

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