4.2 Study of Two-Nucleon Systems with Realistic Interactions
149
Exercise II
Using Eqs. (4.1)–(4.4), show that H 2 = H CM + h rel .
Consequently, one can solve the two-body eigenproblem generated by Eq. (4.6)
using the center-of-mass and relative eigenproblems with associated energies and
wave functions:
H 2 |Ψ = E|Ψ
(4.11)
|Ψ = |Ψ rel CM
(4.12)
E = E CM + e rel
(4.13)
H CM |Ψ CM = E CM |Ψ CM
(4.14)
h rel |Ψ rel = e rel |Ψ rel .
(4.15)
Note that one can add a potential depending on center-of-mass coordinates without
affecting previous results in Eq. (4.9). While this Hamiltonian is not physical, it is
nevertheless very important theoretically when considered in the many-body case
using a harmonic oscillator center-of-mass potential.
The fundamental importance of a harmonic oscillator center-of-mass Hamiltonian is that the decomposition on |Ψ rel and |Ψ CM in Eq. (4.12) can be imposed
also in the many-body case if a basis of harmonic oscillator states is used. For this,
one adds a center-of-mass Hamiltonian H CM to the initial Hamiltonian H , so that the
center-of-mass part of the eigenstates of the Hamiltonian is the ground state of H CM .
Indeed, H CM and H commute in a so-called N ¯
hω finite model space built from
all Slater determinants of energy smaller than N ¯
hω. Consequently, the separation
of Eq. (4.12) can be also exactly obtained in N ¯
hω spaces by projecting |Ψ CM on
the 0s harmonic oscillator ground state of H CM . This is the object of the Lawson
method [1] which is of fundamental importance in shell model and will be discussed
in Chap. 5.
4.2
Study of Two-Nucleon Systems with Realistic Interactions
Physics of a two-particle system is described solely by Eq. (4.15) which describes
internal structure of the system in relative variables. In the following, we will
study three different two-nucleon systems: dineutron, diproton, and deuteron. These
systems are of fundamental importance in nuclear physics. Indeed, they are the
lightest nontrivial nucleonic systems whose binding energy depends on the nuclear
interaction. Moreover, they are convenient to study experimentally via protonproton and electron-deuteron scattering experiments. While proton-proton collisions
give insight to the nuclear interaction when like particles are involved, the electrondeuteron scattering allows to investigate meson exchange process, which is the main
component of the nuclear interaction. Despite complicated nature of the nuclear
149
Exercise II
Using Eqs. (4.1)–(4.4), show that H 2 = H CM + h rel .
Consequently, one can solve the two-body eigenproblem generated by Eq. (4.6)
using the center-of-mass and relative eigenproblems with associated energies and
wave functions:
H 2 |Ψ = E|Ψ
(4.11)
|Ψ = |Ψ rel CM
(4.12)
E = E CM + e rel
(4.13)
H CM |Ψ CM = E CM |Ψ CM
(4.14)
h rel |Ψ rel = e rel |Ψ rel .
(4.15)
Note that one can add a potential depending on center-of-mass coordinates without
affecting previous results in Eq. (4.9). While this Hamiltonian is not physical, it is
nevertheless very important theoretically when considered in the many-body case
using a harmonic oscillator center-of-mass potential.
The fundamental importance of a harmonic oscillator center-of-mass Hamiltonian is that the decomposition on |Ψ rel and |Ψ CM in Eq. (4.12) can be imposed
also in the many-body case if a basis of harmonic oscillator states is used. For this,
one adds a center-of-mass Hamiltonian H CM to the initial Hamiltonian H , so that the
center-of-mass part of the eigenstates of the Hamiltonian is the ground state of H CM .
Indeed, H CM and H commute in a so-called N ¯
hω finite model space built from
all Slater determinants of energy smaller than N ¯
hω. Consequently, the separation
of Eq. (4.12) can be also exactly obtained in N ¯
hω spaces by projecting |Ψ CM on
the 0s harmonic oscillator ground state of H CM . This is the object of the Lawson
method [1] which is of fundamental importance in shell model and will be discussed
in Chap. 5.
4.2
Study of Two-Nucleon Systems with Realistic Interactions
Physics of a two-particle system is described solely by Eq. (4.15) which describes
internal structure of the system in relative variables. In the following, we will
study three different two-nucleon systems: dineutron, diproton, and deuteron. These
systems are of fundamental importance in nuclear physics. Indeed, they are the
lightest nontrivial nucleonic systems whose binding energy depends on the nuclear
interaction. Moreover, they are convenient to study experimentally via protonproton and electron-deuteron scattering experiments. While proton-proton collisions
give insight to the nuclear interaction when like particles are involved, the electrondeuteron scattering allows to investigate meson exchange process, which is the main
component of the nuclear interaction. Despite complicated nature of the nuclear