2.2 Quantum Elastic Scattering
57
Equating (2.56) to (2.68) and using the asymptotic expression of the plane wave
given in Eq. (2.64), canceling the incoming wave (i.e. the coefficients of the term
e
−ikr ) allows us to determine coefficients A l (the calculation is a bit long but not
difficult):
A l = (2l + 1)i
l e
iδ l /k.
(2.69)
With simple algebra,
17 we can then write the full asymptotic form of the wave function
(see Eq. 2.68)
(r)
r→∞
∼
∞
l=0
(2l + 1)i
l e
iδ l
sin(kr − lπ/2 + δ l )
k
P l (cos θ)
(2.70)
=
1
2ikr
∞
l=0
(2l + 1)i
l e
iδ l
e
i(kr−lπ/2+δ l )
− e
−i(kr−lπ/2+δ l )
P l (cos θ)
=
1
2ikr
∞
l=0
(2l + 1)
S l − (−1)
l e
−ikr
P l (cos θ)
where S l = e
2iδ l is the element of the S matrix
18 in the case of elastic scattering.
From Eq. 2.57, we obtain
19 :
f (θ) =
2i
k
∞
l=0
A l e
iδ l − ˜
A l
e
−ilπ/2 P l (cos θ)
(2.71)
=
1
2ik
∞
l=0
e
−ilπ/2
(2l + 1)i
l
[S l − 1]P l (cos θ)
=
1
2ik
∞
l=0
(2l + 1)[S l − 1]P l (cos θ)
which gives us the scattering amplitude as a function of the phase shift δ l . Taking
into account that the Legendre polynomials satisfy the relationship
20
17 From the formulae sin z =
e iz −e −iz
2i
, also e −lπ/2 = (1/i) l , and by definition of complex number
i 2l ≡ (−1) l .
18 The scattering matrix S plays a fundamental role (see text [5, 8]) in the scattering theory because
it is the quantity linking theory and experiment.
19 In the penultimate of the following steps, we use the trivial transformation
e 2iδ l −1
2i
=
e iδ l (e iδ l −e −iδ l )
2i
= e iδ l sin δ l .
20 Derived from the orthonormality relationship of the Legendre polynomials, see what has been
said in Appendix A3.
57
Equating (2.56) to (2.68) and using the asymptotic expression of the plane wave
given in Eq. (2.64), canceling the incoming wave (i.e. the coefficients of the term
e
−ikr ) allows us to determine coefficients A l (the calculation is a bit long but not
difficult):
A l = (2l + 1)i
l e
iδ l /k.
(2.69)
With simple algebra,
17 we can then write the full asymptotic form of the wave function
(see Eq. 2.68)
(r)
r→∞
∼
∞
l=0
(2l + 1)i
l e
iδ l
sin(kr − lπ/2 + δ l )
k
P l (cos θ)
(2.70)
=
1
2ikr
∞
l=0
(2l + 1)i
l e
iδ l
e
i(kr−lπ/2+δ l )
− e
−i(kr−lπ/2+δ l )
P l (cos θ)
=
1
2ikr
∞
l=0
(2l + 1)
S l − (−1)
l e
−ikr
P l (cos θ)
where S l = e
2iδ l is the element of the S matrix
18 in the case of elastic scattering.
From Eq. 2.57, we obtain
19 :
f (θ) =
2i
k
∞
l=0
A l e
iδ l − ˜
A l
e
−ilπ/2 P l (cos θ)
(2.71)
=
1
2ik
∞
l=0
e
−ilπ/2
(2l + 1)i
l
[S l − 1]P l (cos θ)
=
1
2ik
∞
l=0
(2l + 1)[S l − 1]P l (cos θ)
which gives us the scattering amplitude as a function of the phase shift δ l . Taking
into account that the Legendre polynomials satisfy the relationship
20
17 From the formulae sin z =
e iz −e −iz
2i
, also e −lπ/2 = (1/i) l , and by definition of complex number
i 2l ≡ (−1) l .
18 The scattering matrix S plays a fundamental role (see text [5, 8]) in the scattering theory because
it is the quantity linking theory and experiment.
19 In the penultimate of the following steps, we use the trivial transformation
e 2iδ l −1
2i
=
e iδ l (e iδ l −e −iδ l )
2i
= e iδ l sin δ l .
20 Derived from the orthonormality relationship of the Legendre polynomials, see what has been
said in Appendix A3.