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8.1. Coulomb scattering of charged spin-0 particles
where
μ
′ ∗
′ ∗
j
(x) = ie(φ ∂
μ φ − (∂
μ φ )φ)
(8.10)
em,s +
can be regarded as an electromagnetic ‘transition current’, analogous to the
simple probability current for a single state. In the following section we shall
see the exact meaning of this idea, using quantum field theory. Meanwhile,
′
we insert the plane-wave free-particle solutions (8.6) and (8.7) for φ and φ
into (8.10) to obtain
μ
′ )
μ −i(p−p )·x
j
(x) = N N
′ e(p + p e
'
(8.11)
em,s +
so that (8.9) becomes
∫
−i(p−p
A s + = −iN N
′
d
4 x e(p + p
′ ) μ e
' )·x A
μ (x).
(8.12)
In the case of Coulomb scattering from a static point charge Ze (e > 0),
the vector potential A
μ is given by
A
0 =
Ze
4π|x|
A = 0.
(8.13)
Inserting (8.13) into (8.12) we obtain
A s + = −iN N
′ Ze
2 (E + E
′ )
∫
e
−i(E−E
' )t dt
∫ e
i(p−p
' )·x
4π|x|
d
3
x.
(8.14)
The initial and final 4-momenta are
′
p = (E, p)
p = (E
′ , p
′ )
√
√
′ 2
with E = M 2 + p 2 , E
′ = M 2 + p . The first (time) integral in (8.14)
gives an energy-conserving δ-function 2πδ(E − E
′ ) (see appendix E), as is
expected for a static (non-recoiling) scattering centre. The second (spatial)
integral is the Fourier transform of 1/4π|x|, which can be obtained from (1.13),
(1.26) and (1.27) by setting m U = 0; the result is 1/q
2 where q = p−p
′ . Hence
Ze
2
A s + = −iN N
′ 2πδ(E − E
′ )
2E
(8.15)
q 2
≡ −i(2π)δ(E − E
′ )V s +
(cf equation (A.25))
(8.16)
where in (8.15) we have used E = E
′ in the matrix element. This is in the
standard form met in time-dependent perturbation theory (cf equations (A.25)
and (A.26)).
The transition probability per unit time is then (appendix H, equation
(H.18))
P ˙ s + = 2π|V s + |
2 ρ(E
′ )
(8.17)
8.1. Coulomb scattering of charged spin-0 particles
where
μ
′ ∗
′ ∗
j
(x) = ie(φ ∂
μ φ − (∂
μ φ )φ)
(8.10)
em,s +
can be regarded as an electromagnetic ‘transition current’, analogous to the
simple probability current for a single state. In the following section we shall
see the exact meaning of this idea, using quantum field theory. Meanwhile,
′
we insert the plane-wave free-particle solutions (8.6) and (8.7) for φ and φ
into (8.10) to obtain
μ
′ )
μ −i(p−p )·x
j
(x) = N N
′ e(p + p e
'
(8.11)
em,s +
so that (8.9) becomes
∫
−i(p−p
A s + = −iN N
′
d
4 x e(p + p
′ ) μ e
' )·x A
μ (x).
(8.12)
In the case of Coulomb scattering from a static point charge Ze (e > 0),
the vector potential A
μ is given by
A
0 =
Ze
4π|x|
A = 0.
(8.13)
Inserting (8.13) into (8.12) we obtain
A s + = −iN N
′ Ze
2 (E + E
′ )
∫
e
−i(E−E
' )t dt
∫ e
i(p−p
' )·x
4π|x|
d
3
x.
(8.14)
The initial and final 4-momenta are
′
p = (E, p)
p = (E
′ , p
′ )
√
√
′ 2
with E = M 2 + p 2 , E
′ = M 2 + p . The first (time) integral in (8.14)
gives an energy-conserving δ-function 2πδ(E − E
′ ) (see appendix E), as is
expected for a static (non-recoiling) scattering centre. The second (spatial)
integral is the Fourier transform of 1/4π|x|, which can be obtained from (1.13),
(1.26) and (1.27) by setting m U = 0; the result is 1/q
2 where q = p−p
′ . Hence
Ze
2
A s + = −iN N
′ 2πδ(E − E
′ )
2E
(8.15)
q 2
≡ −i(2π)δ(E − E
′ )V s +
(cf equation (A.25))
(8.16)
where in (8.15) we have used E = E
′ in the matrix element. This is in the
standard form met in time-dependent perturbation theory (cf equations (A.25)
and (A.26)).
The transition probability per unit time is then (appendix H, equation
(H.18))
P ˙ s + = 2π|V s + |
2 ρ(E
′ )
(8.17)