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32 Chapter 5. The Eikonal Final State
approximation (RDWIA). Next, we will introduce a consistent relativistic model for
the description of A(e, e ′ p) processes in the eikonal approximation. With the term
“consistent” we refer to a procedure in which the bound and scattering states are
derived from the same Dirac equation. The consistent approach possesses the virtue
of obeying orthogonality and unitarity constraints but will soon be observed to
provide poor descriptions of the experimental data. In a next step, we will improve
our model by implementing complex optical potentials for the description of proton
distortion in the final state. Finally, we develop a scheme that allows to perform fully
relativistic and unfactorized A(e, e ′ p) calculations in a Glauber multiple-scattering
framework.
5.1 The Relativistic Distorted Wave Approximation
We will now briefly outline the RDWIA scheme to solve the Dirac equation of
Eq. (3.3). Within a relativistic framework the bound state wave functions with
well defined angular quantum numbers κ and m were of the form
ψ nκm (⃗r) =
[ ]
ıGnκt (r)/r Y κm
−F nκt (r)/r Y −κm
. (5.1)
These solutions are eigenstates of the total Hamiltonian and the total angular momentum
with eigenvalue j = |κ| − 1/2,
Y κm = ∑ 1
< lm l
m l m s
2 m s|l 1 2 jm > Y l,m l
χ 1
2 ms ,
j = |κ| − 1 2 , l = {
κ, κ > 0
−(κ + 1), κ < 0 .
(5.2)
The wave function for the outgoing proton Ψ f is a scattering solution of the same
Dirac equation of Eq. (3.3). In the RDWIA approach it is obtained as a partial wave
expansion in configuration space [17, 18, 52, 53],
Ψ f (⃗r) = 4π
√
E f + M
2M
∑
1
e −ıδ∗ κ
< lm l
κ,m,m l
2 m s|l 1 2 jm > Y lm ∗
l
(ˆk f )ψ κm (⃗r) . (5.3)
The phase shifts δ κ are calculated with a procedure as for example outlined in
Ref. [54]. Basically, one expresses the incident, or bound, wave as a sum of partial
waves and one can then compare the asymptotic radial functions of Eq. (5.3) with
this plane wave expansion to determine the phase-shifts.