252
Chapter 4. Particle scattering
This is the dependence seen by Rutherford in the scattering of
alpha particles by a gold foil, from which the nuclear model of the
atom was originally deduced. It is therefore called the Rutherford
scattering cross section. It is the same for both signs of κ, and is
therefore independent of whether the scattering force is attractive
or repulsive. It approaches infinity at zero scattering angle, and
has a finite value for backscattering at ϑ = π. Integrating over all
solid angle, we form the total cross section. This is
π
σ tot = 2π
σ(ϑ) sin ϑ dϑ.
(4.55)
0
This is infinite for the case of Rutherford scattering. Physically,
this means that the Coulomb potential effectively has infinite
range.
We are now in a good position to consider quantum mechanical
elastic scattering. This is the subject of the next three sections.
4.3 Integral expression of Schr¨ odinger’s
equation
All relevant information about quantum mechanical scattering is
contained in the differential cross section, which was defined in
the preceding section. The cross section in turn depends on the
wave function ψ(x, t), which is a solution of the time-dependent
Sch¨ odinger equation (3.13) with appropriate boundary conditions.
In this section we cast Schr¨ odinger’s equation in a form which will
prove to be directly applicable to the scattering problem.
In the important special case where the electrostatic potential φ(x)
has no explicit time dependence, the wave function ψ(x, t) takes
on the separable form (3.18) where
−iHt/¯ h
ψ(x, t) = u(x) e
,
(4.56)
Chapter 4. Particle scattering
This is the dependence seen by Rutherford in the scattering of
alpha particles by a gold foil, from which the nuclear model of the
atom was originally deduced. It is therefore called the Rutherford
scattering cross section. It is the same for both signs of κ, and is
therefore independent of whether the scattering force is attractive
or repulsive. It approaches infinity at zero scattering angle, and
has a finite value for backscattering at ϑ = π. Integrating over all
solid angle, we form the total cross section. This is
π
σ tot = 2π
σ(ϑ) sin ϑ dϑ.
(4.55)
0
This is infinite for the case of Rutherford scattering. Physically,
this means that the Coulomb potential effectively has infinite
range.
We are now in a good position to consider quantum mechanical
elastic scattering. This is the subject of the next three sections.
4.3 Integral expression of Schr¨ odinger’s
equation
All relevant information about quantum mechanical scattering is
contained in the differential cross section, which was defined in
the preceding section. The cross section in turn depends on the
wave function ψ(x, t), which is a solution of the time-dependent
Sch¨ odinger equation (3.13) with appropriate boundary conditions.
In this section we cast Schr¨ odinger’s equation in a form which will
prove to be directly applicable to the scattering problem.
In the important special case where the electrostatic potential φ(x)
has no explicit time dependence, the wave function ψ(x, t) takes
on the separable form (3.18) where
−iHt/¯ h
ψ(x, t) = u(x) e
,
(4.56)