Probability amplitude, symmetric states

Figure 9. Transition probability amplitude, to- o( ) assisted by the 3-rd effective complex resonance potential energy surface, (a) phase, (b) absolute value, (c) trajectory. The phase changes by tt in resonances. The phase drops by tt between resonances, when t E) intersects the origin. The position of resonance states is plotted in squares (open squares -symmetric resonance states, full squares - antisymmetric resonance states). Note that due to the symmetry properties of the problem, transition is not assisted by antisymmetric resonances.

The form displayed in eq. (2-40) implies that the ratios of the amplitudes for scattering into different exit channels are independent of the entrance channel. This, of course, will only be true if the resonance is long lived, so that memory of the initial state can be lost. Note that Aga is a symmetric function, which is a consequence of time-reversal invariance. Note also that, within the approximations used, the phase shift associated with a given channel is just the elastic scattering phase shift for that channel. Finally, the partial widths are proportional to the probability of decay from channel fi. Equation (2-41) is, then, merely a statement that the total probability of decay from channel is the sum of the probabilities of decay into individual channels. 

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