Chapter 8 Scattering Theory - Particle Physics Group

CHAPTER 8. SCATTERING THEORY 158
Ω † ±Ω ± = ∑ ij
Ω ± Ω † ± = ∑ ij
|u 0 i 〉〈u ± i |u± j 〉〈u0 j| = ∑ ij
|u ± i 〉〈u0 i |u 0 j〉〈u ± j | = ∑ i
|u 0 i 〉δ ij 〈u 0 j| =½
|u ± i 〉〈u± i | =½−P b.s. (8.168)
states, which are not complete if the potential V supports bound states. More explicitly, we
can write
∑
Ω ± = Ω ± |u 0 i 〉〈u 0 i | = ∑ |u ± i 〉〈u0 i |. (8.166)
i
i
Hence
(8.167)
is established.
S † S = Ω † +Ω − Ω † −Ω + = Ω † +(½−P b.s. )Ω + = Ω † +Ω + =½
SS † = Ω † −Ω + Ω † +Ω − = Ω † −(½−P b.s. )Ω − = Ω † −Ω − =½
where P b.s. is the projector to the bound states. If the potential V has negative energy solutions
these states cannot be produced in a scattering process and are hence missing from the
completeness relation in the last sum. Combining these results unitarity of the S matrix
(8.169)
(8.170)
Relating the S-matrix to the transition matrix. In order to derive the relation
between S and T we write the S-matrix elements S ij as
With 〈u + i |u+ j 〉 = δ ij and
we obtain
Since H|u + j 〉 = E j|u + j 〉 and
S ij = 〈u − i |u+ j 〉 = 〈u+ i |u+ j 〉 + (〈u− i | − 〈u+ i |) |u+ j 〉. (8.171)
〈u − i | = 〈u0 i |Ω † − = 〈u 0 1
i |(1 + V ),
E i − H + iε
(8.172)
〈u + i | = 〈u0 i |Ω † + = 〈u 0 1
i |(1 + V ).
E i − H − iε
(8.173)
S ij = δ ij + 〈u 0 1
i |V (
E i − H + iε − 1
E i − H − iε )|u+ j 〉. (8.174)
lim(
ε→0
we conclude S ij = δ ij − 2πiδ(E i − E j )〈u 0 i |V |u + j 〉 and hence
1
z − iε − 1 ) = 2πi δ(z) (8.175)
z + iε
S ij = δ ij − 2πiδ(E i − E j )T ij . (8.176)
The non-relativistic dispersion E = (k) 2 /2m implies δ(E i − E j ) = m
2 k δ(k i − k j ) so that
〈k ′ |S|k〉 = δ 3 ( ⃗ k − ⃗ k ′ ) + i
2πk δ(k − k′ )f( ⃗ k ′ , ⃗ k). (8.177)
Partial wave decomposition on the energy shell [Hittmair] then leads to S l = e 2iδ l = 1 − 2πiTl .