Consistent chiral three-nucleon interactions in ... - Theory Center

3 Mathematical basics
antisymmetrizer in the next subsection and also later in the transformation pre-
sented in section 4. There are two different definitions of HOBs, distinguished by
the order of the quantum numbers in the bra state. We first present the definition
that we use in subsections 3.3.1 and 3.3.2. After that, we introduce the alternative
definition in subsections 3.3.3 and 3.3.4. In subsection 3.3.5 we point out how to
relate the two definitions to each other.
3.3.1 Definition of harmonic oscillator brackets – our version
The definition of the harmonic oscillator brackets that we use is given by
|[n1l1, n2l2]Λλ〉 =
〈(NL, nl)Λ|(n1l1, n2l2)Λ〉d |[NL, nl]Λλ〉 , (3.41)
NL,nl
=
〈〈NL, nl; Λ|n1l1, n2l2〉〉d |[NL, nl]Λλ〉 , (3.42)
NL,nl
with 〈〈NL, nl; Λ|n1l1, n2l2〉〉d as the HOB, where d is a non-negative real number
[20]. The corresponding coordinate transformation matrix is given by the general
form of an orthogonal transformation
d
⎛
R 1+d
= ⎝
r 1
1+d −
1
1+d
d
1+d
⎞
⎠
r1
r2
. (3.43)
For d = 1 this is a transformation from single-particle coordinates to relative and
center-of-mass coordinates. For other values of d we describe a general orthogo-
nal transformation. Although it might in general be a bit misleading, we refer to N
and L as the quantum numbers of the “center-of-mass” of the two particles, and to
n and l as the quantum numbers of their “relative” motion. Here N and n are the
radial quantum numbers of the harmonic oscillator. With the notation introduced
in section 3.2 we should write L( R), l(r) and accordingly l1(r1) and l2(r2) for the dif-
ferent angular momentum quantum numbers. For brevity we drop this notation
for the rest of this subsection. The quantum number Λ is the coupled orbital angu-
lar momentum of the two particles and λ the corresponding projection quantum
number. If we want to calculate the HOB explicitly, we insert an identity operator
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