We begin by constructing all possible higher-order and higher-rank tensor
operators from powers of the derivative
, where
are the spin-projection components of the vector (rank-1)
operator
. It is obvious that all possible
th-order powers
of the derivative can be written as sums of terms
. Therefore, any
th-order
power is simply obtained by multiplying some
th-order power by
a sum of
operators. Then, powers of a given rank can be
obtained iteratively by vector coupling.
In the second order,
the two nabla operators can be coupled to angular momenta 0
and 2. The coupling to angular momentum 0,
, corresponds to the Laplacian operator.
Furthermore, the coupling to angular momentum 2,
, gives
the second-order, rank-2 derivative operator. The
rank-1 coupling,
, vanishes because the
derivatives commute. Similarly, in each one higher order, a rank-
symmetric operator can be coupled with
only to
and
. Hence, all the
th-order powers have the form of
multiplied by the
th-order rank-
(stretched)
coupled operators for
. Then, up to N
LO,
one obtains 16 different operators
listed in Table 1.
Any arbitrary tensor formed by coupled operators
can always be rewritten as a sum of operators
through the repeated use of the
symbols.
Exactly in the same way, we define 16 different operators ,
which are spherical tensors built of the relative momentum operators
coupled up to N
LO, i.e., for
and
. In the
remainder of this section, we only discuss operators
, while
all the results mutatis mutandis also pertain to operators
.
The stretched coupled operators for
,
![]() |
(11) |
Equivalently, derivative operators
can be written in the Cartesian representation, in which their
components are numbered by
Cartesian indices,
,
. The order of these indices does not matter (totally symmetric
tensors) and all traces vanish,
![]() |
(12) |
We note here in passing that we could have equally well used the
Cartesian derivative operators with traces not subtracted out, i.e.,
In principle, below one could replace the spherical representations of derivative operators shown in Table 1 by their Cartesian counterparts (13)-(17) or (18)-(22), and work entirely in the Cartesian representation. However, in our opinion, the use of the spherical representation is superior and more economical. Moreover, whenever calculation of the Cartesian derivatives is more suitable, we may express spherical components of the derivative operators through the Cartesian derivatives, as shown in Table 2. An example of using the Cartesian representation (18)-(22) is given in Sec. 4.