ABSTRACT
In this chapter, we shall focus on deriving the connection between the scattering matrix (or the S-matrix) and the transition matrix (or the T-matrix) by means of the formalism of the Abel strong limits from chapter 7. As is well-known [7], the Sˆ−operator is defined as the product of the two Møller wave operators according to
Sˆ = Ωˆ−†Ωˆ+
= Lim t→+∞ Limt′→−∞ Ωˆ
−†(t)Ωˆ+(t′)
= Lim t→+∞ Limt′→−∞ {Uˆ
† 0(t)Uˆ(t)} {Uˆ†(t′ )Uˆ0(t′ )}
= Lim t→+∞ Limt′→−∞ {e
∴ Sˆ = Lim t→+∞ Limt′→−∞ {e
iHˆ0t e−iHˆt} {eiHˆt′ e−iHˆ0t′}. (8.1)
The order of the appearance of the two limits in (8.1) is irrelevant. This occurs because in the definition (8.1) of the stationary Møller wave operators Ωˆ± = Lim
τ→∓∞{Uˆ †(τ)Uˆ0(τ)} , we set τ = t and τ = t′ for Ωˆ− and Ωˆ+, respectively.
However, time τ appears as a dummy variable and, as such, associating τ with t or t′ is arbitrary. In other words, we could have made the opposite choice such that τ = t′ and τ = t correspond to Ωˆ− and Ωˆ+, respectively
