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3.2. Results 19
have verified that these results are comparable to those produced by the TIMORA
code [16], which is widely used to solve the set of Eq. (3.7).
The scalar potential is entirely governed by the scalar meson field (σ meson),
and is responsible for the attractive force that keeps the nucleons bound within the
nucleus. The repulsive vector potential has contributions from the vector (ω and
ρ) meson fields and the Maxwell field. It should be noted that the contributions
from the ρ meson field and the Maxwell field are several orders of magnitude smaller
than the contribution from the ω meson field. Since the potential experienced by
the neutrons differs only from the one experienced by the protons in these two
minor terms, the total scalar and vector potentials for protons and neutrons are
almost exactly the same. That is why we have only depicted the proton potentials
in Fig. 3.2. These very same potentials will be used in Sec. 5 to derive the eikonal
scattering states.
A point of high interest in nuclear physics are the nucleon spectral functions
and momentum distributions. In calculating these contributions, we closely follow
the method outlined in Ref. [38]. In free space the nucleon’s motion is governed
by the free particle solution of the Dirac equation. These plane wave states with
momentum ⃗ k and energy ɛ k = √ k 2 + M 2 are given by
φ ⃗k,s (⃗r) =
√ [
ɛ k + M
2M
1
E+M
]
1
⃗σ · ⃗k
e ı⃗k·⃗r χ 1 . (3.10)
ms
2
Since the Dirac-Hartee potentials are spherically symmetric, we expand this free
space solution in terms of the spin-spherical harmonics of Eq. (3.6) :
φ ⃗k,s (⃗r) =
×
√
ɛ k + M ∑
4π(+ı) l 1
< lm l
2M
κmm l
2 m s|l 1 2 jm > Y lm ⋆
l
(Ω ⃗k )
kr
[ ]
ıgκ (kr) Y κm
, (3.11)
−f κ (kr) Y −κm
where the usual spherical harmonics Y lml are given in terms of the associated Legendre
polynomials P m l
l
:
√ ( )
(2l + 1) (l − ml )!
Y lml (θ, φ) =
(−1) m l
e ımlφ P m l
l
(cos θ) . (3.12)
4π (l + m l )!
The radial components of the wave function are written in terms of Ricatti-Bessel
functions (these are the spherical Bessel functions times their argument) :
g κ (kr) = ĵ l (kr)
f κ (kr) =
±k
ɛ k + M ĵl∓1(kr) , (3.13)