62
2 The Quantum Approach to the Two-Body Problem
where δ l is a phase shift formulated as
δ l = arg(l + 1 − i/k).
(2.83)
Comparing (2.82) with its asymptotic form, we see that the only difference lies in the
logarithmic term in the argument of the sin function at the numerator. In fact, in the
absence of a Coulomb field,
22
δ l = arg [(l + 1 − i/k)] = 0. As a result Eq. (2.82)
coincides with the partial wave expansion of the plane wave (see Eq. 2.64).
To the end of calculating the scattering amplitude (and therefore the cross section)
we get from Eq. 2.83
e
2iδ l ≡ S l =
(l + 1 + i/k)
(l + 1 − i/k)
(2.84)
and obtain the resulting scattering amplitude is (2.73)
f (θ) =
1
2ik
∞
l=0
(2l + 1)S l P l (cos θ)
(2.85)
=
1
2ik
∞
l=0
(2l + 1)
(l + 1 + i/k)
(l + 1 − i/k)
P l (cos θ)
(2.86)
By definition of the differential cross section (Eq. 1.53) in atomic units, we have
dσ
d
= |f (θ)|
2
=
1
4k 4 [arcsin(θ/2)]
4
(2.87)
that coincides with the Rutherford formula (1.65) of the classical treatment. Let us
not forget the laboratory process by which we obtained this especially interesting
fact that the solution is really asymptotic only at large distances.
2.3.2 The Rigid Sphere
As just shown, the apparent simplicity of the bound state solution of the attractive
Coulomb potential is clearly opposed to the complexity of its scattering solution. A
better model to choose in order to find more intuitive connections between the shape
of the interaction and the formulation of scattering quantities and to compare as well
their classical and quantum solution is the rigid sphere one. The rigid sphere model
is a pure scattering case (as opposed to the HO one that is exclusively a pure bound
22 In fact, a property of the special function is
(l + 1 − i/k) = (l − i/k) · · · (1 − i/k))(1 − i/k)
.
2 The Quantum Approach to the Two-Body Problem
where δ l is a phase shift formulated as
δ l = arg(l + 1 − i/k).
(2.83)
Comparing (2.82) with its asymptotic form, we see that the only difference lies in the
logarithmic term in the argument of the sin function at the numerator. In fact, in the
absence of a Coulomb field,
22
δ l = arg [(l + 1 − i/k)] = 0. As a result Eq. (2.82)
coincides with the partial wave expansion of the plane wave (see Eq. 2.64).
To the end of calculating the scattering amplitude (and therefore the cross section)
we get from Eq. 2.83
e
2iδ l ≡ S l =
(l + 1 + i/k)
(l + 1 − i/k)
(2.84)
and obtain the resulting scattering amplitude is (2.73)
f (θ) =
1
2ik
∞
l=0
(2l + 1)S l P l (cos θ)
(2.85)
=
1
2ik
∞
l=0
(2l + 1)
(l + 1 + i/k)
(l + 1 − i/k)
P l (cos θ)
(2.86)
By definition of the differential cross section (Eq. 1.53) in atomic units, we have
dσ
d
= |f (θ)|
2
=
1
4k 4 [arcsin(θ/2)]
4
(2.87)
that coincides with the Rutherford formula (1.65) of the classical treatment. Let us
not forget the laboratory process by which we obtained this especially interesting
fact that the solution is really asymptotic only at large distances.
2.3.2 The Rigid Sphere
As just shown, the apparent simplicity of the bound state solution of the attractive
Coulomb potential is clearly opposed to the complexity of its scattering solution. A
better model to choose in order to find more intuitive connections between the shape
of the interaction and the formulation of scattering quantities and to compare as well
their classical and quantum solution is the rigid sphere one. The rigid sphere model
is a pure scattering case (as opposed to the HO one that is exclusively a pure bound
22 In fact, a property of the special function is
(l + 1 − i/k) = (l − i/k) · · · (1 − i/k))(1 − i/k)
.