H
Elements of Non-relativistic Scattering
Theory
H.1 Time-independent formulation and differential cross
section
We consider the scattering of a particle of mass m by a fixed spherically
symmetric potential V (r); we shall retain ħ explicitly in what follows. The
potential is assumed to go to zero rapidly as r → ∞, as for the Yukawa
potential (G.16); it will turn out that the important Coulomb case can be
treated as the a → ∞ limit of (G.16). We shall treat the problem here as a
stationary state one, in which the Schr¨ odinger wavefunction ψ(r, t) has the
form
ψ(r, t) = φ(r)e
−iEtħ
(H.1)
where E is the particle’s energy, and where φ(r) satisfies the equation
[
]
−ħ
2
∇
2 + V (r) φ(r) = Eφ(r).
(H.2)
2m
We shall take V to be spherically symmetric, so that V (r) = V (r) where
r = |r|. In this approach to scattering, we suppose the potential to be ‘bathed’
in a steady flux of incident particles, all of energy E. The wavefunction for
the incident beam, far from the region near the origin where V is appreciably
non-zero, is then just a plane wave of the form φ inc = e
ikz , where the z-axis
has been chosen along the propagation direction, and where E = ħ
2
k
2 /2m
with k = (0, 0, k). This plane wave is normalized to one particle per unit
volume, and yields a steady-state flux of
ħ
j inc =
[φ inc
∗
∇φ inc − φ inc ∇φ
∗
inc ]
2mi
= ħk/m = p/m
(H.3)
where the momentum is p = ħk. As expected, the incident flux is given by
the velocity v per unit volume.
Though we have represented the incident beam as a plane wave, it will,
in practice, be collimated. We could, of course, superpose such plane waves,
399
Elements of Non-relativistic Scattering
Theory
H.1 Time-independent formulation and differential cross
section
We consider the scattering of a particle of mass m by a fixed spherically
symmetric potential V (r); we shall retain ħ explicitly in what follows. The
potential is assumed to go to zero rapidly as r → ∞, as for the Yukawa
potential (G.16); it will turn out that the important Coulomb case can be
treated as the a → ∞ limit of (G.16). We shall treat the problem here as a
stationary state one, in which the Schr¨ odinger wavefunction ψ(r, t) has the
form
ψ(r, t) = φ(r)e
−iEtħ
(H.1)
where E is the particle’s energy, and where φ(r) satisfies the equation
[
]
−ħ
2
∇
2 + V (r) φ(r) = Eφ(r).
(H.2)
2m
We shall take V to be spherically symmetric, so that V (r) = V (r) where
r = |r|. In this approach to scattering, we suppose the potential to be ‘bathed’
in a steady flux of incident particles, all of energy E. The wavefunction for
the incident beam, far from the region near the origin where V is appreciably
non-zero, is then just a plane wave of the form φ inc = e
ikz , where the z-axis
has been chosen along the propagation direction, and where E = ħ
2
k
2 /2m
with k = (0, 0, k). This plane wave is normalized to one particle per unit
volume, and yields a steady-state flux of
ħ
j inc =
[φ inc
∗
∇φ inc − φ inc ∇φ
∗
inc ]
2mi
= ħk/m = p/m
(H.3)
where the momentum is p = ħk. As expected, the incident flux is given by
the velocity v per unit volume.
Though we have represented the incident beam as a plane wave, it will,
in practice, be collimated. We could, of course, superpose such plane waves,
399