ABSTRACT
From the properties (5.11), (5.12) and (5.16) of the definition of a quantum collision system, it can be concluded that the correspondence between ψ ∈ H and ψ+ = Ωˆ+ψ ∈ R represents a linear, isometric mapping of the sub-space H to R. Linearity is proven as follows
Lim t→∓∞ Uˆ
†(t)Uˆ0(t)(λ1ψ1 + λ2ψ2) = λ1 Lim t→∓∞ Uˆ
+ λ2 Lim t→∓∞ Uˆ
∴ Lim t→∓∞ Uˆ
†(t)Uˆ0(t)(λ1ψ1 + λ2ψ2) = λ1ψ±1 + λ2ψ ± 2 (10.1)
where λ1, λ2 ∈ C and ψ1, ψ2 ∈ H. Isometry means that the mapping preserves the norm as stated via
||ψ|| = ||ψ±|| = ||Ωˆ±ψ||. (10.2)
Since Uˆ0(t) and Uˆ(t) are isometric operators, we have
||ψ±||2 = 〈ψ±|ψ±〉 = 〈 Lim
t→∓∞ Uˆ †(t)Uˆ0(t)ψ| Lim
t→∓∞ Uˆ †(t)Uˆ0(t)ψ〉
= Lim t→∓∞ 〈Uˆ
†(t)Uˆ0(t)ψ|Uˆ†(t)Uˆ0(t)ψ〉 = Lim
†(t)Uˆ0(t)ψ〉 = Lim
= Lim t→∓∞ 〈ψ|ψ〉 = ||ψ||
∴ ||ψ±||2 = ||ψ||2 (QED). (10.3)
ION-ATOM
This proof can also be accomplished by using isometry of the operator Ωˆ i.e. Ωˆ†Ωˆ = 1ˆ, so that
||Ωˆψ||2 = 〈Ωˆψ|Ωˆψ〉 = 〈ψ|Ωˆ†Ωˆψ〉 = 〈ψ|ψ〉 = ||ψ|| (QED) (10.4) where Ωˆ stands for both Ωˆ±.
