Spherical tensor operators
– see the following reference – an appendix to the book:
Computational Atomic Structure: An MCHF Approach
Charlotte Froese Fischer Vanderbilt University, Tomas Brage and Per Jönsson Lund
University
http://atoms.vuse.vanderbilt.edu/Elements/CompMeth/HF/node24.html
Angular momentum operators can not only be used to classify wavefunctions, but also
operators acting on these functions. If T denotes a general operator acting on the space
spanned by the eigenfunctions of the angular momentum operator , then a spherical
tensor operator of rank k
can be defined as a set of 2k+1 operators,
, that fulfills the following commutation relations
(116)
where
(117)
As is easily verified, the renormalized spherical harmonics
(118)
satisfy
(119)
and can thus be regarded as spherical tensor operators of rank k.
Another important example is given by the vector operator . According to definition,
the Cartesian components of a vector operator transform as the spatial co-ordinates x, y
and z during a rotation. From these transformation properties it can be shown that the
spherical components
V(1)1 =
(120)
V(1)1 = Vz
V(1)-1 =
define a spherical tensor operator
of rank 1. Specifically, the angular momentum
operators themselves may by written as spherical tensor operators
of rank 1.
Scalar operators
According to the definition, an operator T that commutes with all components of the
angular momentum operator is a scalar, or rank zero, operator. From the relations
eq:tensor_eq it is easy to show that the matrix elements of T are diagonal in j and m
(121)
In addition the matrix elements are independent of the m quantum number.
Coupling of tensor operators
Two spherical tensor operators
and
acting on the space spanned by the
eigenfunctions of may be coupled to a spherical tensor operator
the Clebsch-Gordan expansion
of rank K using
(124)
If the two tensor operators
and
have the same rank k1=k2=k they can be
combined to form a scalar tensor, i.e. a tensor of rank zero
(125)
Traditionally, the scalar product of two tensor operators is defined by
(126)
and thus
(127)
When k=1 the tensor scalar product reduces to the ordinary cartesian scalar product
(128)
The Wigner-Eckart theorem
From the commutation relations eq:tensor_eq it can be shown that the matrix element
is proportional to a 3j-symbol
(129)
This relation is known as the Wigner-Eckart theorem. The coefficient
is
called the reduced matrix element of
and is independent of the magnetic quantum
numbers m, m' and q.
From the properties of the 3j-symbol we obtain the following selection rules for the
quantum numbers that have to be fulfilled for the matrix element
be non-zero.
to
m=q+m'
(130)
and
(131)
As an example we consider the matrix elements of the renormalized spherical harmonic
From the Wigner-Eckart theorem we obtain
(132)
Noting that
(133)
it is seen that
(134)
Another important example is the matrix elements of the angular momentum operator .
For the j(1)0 = jz component we have
(135)
Thus,
(136)
and
(137)
Matrix elements of tensor operators between coupled
functions
We will now evaluate matrix elements of the coupled tensor operator
eq:couptens.
In most cases the tensor operators
and
act on different spaces, spanned by
and
, respectively, and so we need to consider matrix elements
between coupled functions
(138)
In this case
=
From this relation several special cases can be obtained. Setting K=0 and k1=k2=k and
using the properties of the 3j- and 9j-symbols it can be shown that
Other special cases of equation eq:tens.pr are obtained by setting k2=0 and k1=0,
respectively and using the properties of the 9-j symbol
Setting k=0, we obtain the reduced matrix element form of the uncoupling formulas
eq:unc1 and eq:unc2 for the scalar operator.
Coupling of two angular momenta
Two angular momentum operators and acting on different parts of a system, for
example two different electrons or, respectively, the space- and spin-part of the electron
wave function, can be coupled to form an angular momentum operator
(103)
for the total system. The operators
observables. The simultaneous eigenfunctions
can be obtained as a linear combination
and jz define a complete set of commuting
corresponding to this set
where the coefficients
are known as vector-coupling, or ClebschGordan, coefficients. With the normal phase conventions the the Clebsch-Gordan
coefficients are real and define a unitary matrix. Thus we have the following
orthogonality relations
(104)
and
(105)
In addition, the Clebsch-Gordan, coefficients have the following important symmetry
property
(106)
Often the Clebsh-Gordan coefficients are given in terms of the more symmetric 3jsymbols
(107)
where we have introduced the notation
(108)
The general 3j-symbol
(109)
has the symmetry properties
(110)
and is nonzero only if
(111)
Coupling of three- and four angular momenta
Three angular momenta can be coupled together in two different ways that are grahically
represented in fig:coupling. Corresponding to these couplings there are two sets of
eigenfunctions
and
where J12 and J23 are intermediate quantum numbers.
Figure: Coupling of three angular momenta
Both these sets are complete, and thus there exists a transformation
(112)
Here the summation need only to be done over J12, since all other quantum numbers are
common to the two sets of eigenfunctions. The coefficients are called recoupling
coefficients and are often given in terms of 6-j symbols
=
(113)
When dealing with four angular momenta, there is a whole set of different coupling
schemes. Two possibilities are defined by the following sets of eigenfunctions
and
The transformation between these two sets is given by
(114)
In this case the coefficients are given in terms of 9-j symbols
=
(115)
where is the sum of all nine j quantum numbers. The 3-,6- and 9-j symbols are
available from a number of tables, for example Rotenberg et. al. (1959).
Computational Atomic Structure:
An MCHF Approach
Charlotte Froese Fischer Vanderbilt University
Tomas Brage and Per Jönsson Lund University
http://atoms.vuse.vanderbilt.edu/Elements/CompMeth/HF/book.html
angular momentum
http://atoms.vuse.vanderbilt.edu/Elements/CompMeth/HF/node24.html