Scattering outgoing states

The initial state of the system is described by a normalized Gaussian wave packet of outgoing scattering states ... 

In accordance with Eqs. (30 and 31), the incoming and outgoing scattering states of the d-dimensional reactive system at energy E are given by... 

The scattering states fulfilling outgoing (incoming) boundary conditions, i.e., which correspond to wave packets controlled in the past (future), are given by... 

The energy and state resolved tiansition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves [F)... 

Scattering wavefunctions represent states with unit incoming flux in one particular channel, n, and outgoing flux in all vibrational states n. The probability for a transition from initial channel n to final channel n is given by Sn n 2, which explains the name scattering matrix. 

The form of Eq. (2.58) appears more complicated than that of Eq. (2.57) because the (E, m 0 vP (f) amplitude is composed of contributions from all degenerate n 0) states. Why use the outgoing states at all then The reason is that in " ordinary scattering events (e.g., the collision of two particles) we use states whose s S > paktis well known to us. These are the outgoing states because when t —> — oo it is ". " fhe contour on the semicircle in the upper half of the complex E plane that vanishes.. .. Thus, supplementing the real E integration by such a contour keeps the E — E — ie... 

The above results deal with photodissociation. A similar formulation can be applied to an inelastic scattering event, for example, the scattering of A + BC from an initial state E, m 0) into a final state E, n 0) of A + BC. This problem can be phrased as our asking for the amplitude of making a transition from a free state E, m 0) at t —> —oo to free state E, n 0 at t — oo. Since E, m+) is known to have evolved from E, m 0) and E, n ) is known to evolve to state E, n 0), the answer is given by the expansion coefficients of an outgoing state in terms of the incoming states. That is,... 

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