Download Fano-Racah Tensorial Algebra

Tutorial on Fano-Racah tensorial algebra
I. Introduction.
The Fano-Racah algebra is usually viewed as a technique to efficiently compute quantities
that arise in applications of quantum mechanics to physical systems. It is indeed this, but
it is in fact much more. As the book by Fano and Racah emphasizes, the subject treats
irreducible sets generally, and these occur in all domains of physics. For example, the
dielectric tensor ij has irreducible components who physical effects are charactaristic of the
rank k of the irreducible set to which the components belong. For this reason we view the
subject of irreducible sets as having broad implications for all of physics. This is particularly
true when we express the results of experimental measurements in terms of an underlying
microscopic physical theory.
Clearly, measurements, since they relate to the geometrical positions of measuring devices,
have simple geometrical transformation properties. It is desirable to take advantage of these
properties in the mathematical notations and operations employed in the description of
the measurments. At the elementary level this is done in classical mechanics by introducing
vector quantities A, B, v, etc. rather than their individual components vx , vy , vz = v1 , v2 , v3 .
The individual components have no meaning unless a coordinate system is given, while the
vector v does, and may be expressed in many ways. Thus if I say the wind is blowing in
that direction at 10 mi/hr and point with my arm I have conveyed the physical concept of
wind velocity.
This concept is independent of a coordinate system, and the entity ”wind velocity” is
treated as a whole, represented mathematically by the symbol ”v”. These symbols generalize to the concept of a tensor T of any arbitrary rank, with components Tijk... that
transform under coordinate transformations as the product of coordinates xi xj xk .... This
is the conventional generalization of tensor analysis, but we regard this generalization as
inadequate in that it fails to give primacy of place to irreducible tensors. Such tensors are
charactarized by a rank k and denoted by Tk . The components of such tensors are denoted
by Tqk . Under coordinate transformations only the components of irreducible tensors of the
same rank intermix:
0k
Tq =
k
X
Tqk0 Dqk0 q (α, β, γ)
q 0 =−k
1
(1)
Indeed irreducible tensorial sets are defined by Eq.(1), where the matrix elements Dqk0 q are
standard representations of rotation transformations. It is the entities Tk that generalize
the familiar vector analysis most fruitfully.
Associated with vector symbols are mathematical operations that have proven their worth
over years of development. Amoung them are the operations of dot-product and crossproduct:
s = a·b
(2)
c = a × b.
(3)
One objective of the algebra of irreducible sets is to generalize these simple operations.
Note that Eq.(2) represents the coupling of two vectors to produce a scaler while two vectors
are coupled to produce a third vector in Eq.(3).
These operations generalize via the coupling of two irreducible tensors Tk1 and Sk2 to
produce another irreducible tensor of rank k3 . We represent this tensor by [Tk1 ×Sk2 ]k3
which reads as ”the tensor Tk1 of rank k1 and the tensor Sk2 of rank k2 are coupled to form
a tensor of rank k3 ”. In terms of tensor components this is written
[Tk1 ×Sk2 ]kq33 =
X
Tqk1 Sqk22 (k1 q1 k2 q2 |k3 q3 )
(4)
q 1 q2
where (k1 q1 k2 q2 |k3 q3 ) represents a set of numbers called Clebsh-Gordan coefficients. Just as
we employ Eqs.(2) and (3) of vector calculus without knowing the corresponding numerical
coefficients, so we seek a notation which employs the coupling of two tensors of arbitrary
rank without explicit use of Clebsh-Gordan coefficients. This algebra is referred to as the
algebra of irreducible tensorial sets. It is developed so as to maximize the correspondence
with ordinary vector calculus.
The equations of vector calculus include the familiar relations
a · [b × c] = a × b · c.
a × [b × c] = b(a · c) − c(a · b).
(4)
(5)
These are recognized as recoupling transformations. In Eq.(5) a scaler formed from three
vectors according to the coupling scheme ”b is coupled to c to form a vector which is coupled
to a to produce a scaler” is related to the scaler ”a coupled to b to form a vector which
is coupled to c” to form a scaler. More generally, such a recoupling transformation where
2
an irreducible tensor formed according to one coupling scheme is related to the same tensor
formed according to an alternate coupling scheme involves a sum of terms as in Eq. (5).
The Fano-Racah algebra generalizes the recoupling schemes familiar from vector analysis
to write general recoupling transformations. While these can be of arbitrary complexity
two generic types are most useful. The first and simplest of these is the recoupling of three
tensors analogous to the recoupling of three vectors. We write
[[Tk1 ×Sk2 ]k12 ×Rk3 ]k4 =
X
(6)
((k1 k2 )k12 , k3 , k4 |k1 , (k2 k3 )k23 , k4 ) × [Tk1 ×[Sk2 ×Rk3 ]k23 ]k4
(7)
k23
where the symbol ((k1 k2 )k12 , k3 , k4 |k1 , (k2 k3 )k23 , k4 ) is a recoupling coefficient. It is a number
that depends upon the values of 6 tensorial ranks k1 , k2 , k3 , k4 , k12 , k23 and is usually written
in terms of the 6-j symbols defined according to




 j1 j2 j12 
= [(2j12 + 1)(2j2 3 + 1)]−1/2 (−1)j1 +j2 +j3 +J


 j3 J j23 
((j1 j2 )j12 , j3 , J|j1 , (j2 j3 )j23 , J)
(8)
Explicit formula for the 6-j symbols have been derived, and their values are easily computed.
The next recoupling transformation that we will often employ consid-
ers the recoupling of four irreducible tensors.
Thus we consider the coupling scheme
(k1 k2 )k12 , (k3 k4 )k34 , k5 representing the coupling of four irreducible tensors and rewrite it
in terms of the alternative coupling (k1 k3 )k13 , (k2 k4 )k24 , k5 of the same four tensors. We
have in the obvious notation:
[[Tk1 ×Sk2 ]k12 × [Rk3 ×Qk4 ]k34 ]k5 =
X
k13 k24
((k1 k2 )k12 , (k3 k4 )k34 , k5 |(k1 k3 )k13 , (k2 k4 )k24 , k5 )[[Tk1 ×Rk3 ]k13 × [Sk2 ×Qk4 ]k24 ]k5
(9)
The recoupling coefficients now depend upon 9 tensorial ranks and are usually given in
terms of the 9-j symbols according to.



k1





k2 k12 



−1/2
k3 k4 k34  = [(2k12 + 1)(2k34 + 1)(2k13 + 1)(2k24 + 1)]







k

13 k24 k5
((k1 k2 )k12 , (k3 k4 )k34 , k5 |(k1 k3 )k13 , (k2 k4 )k24 , k5 )
3
(9)
Again explicit expressions and computer programs are available for the computation of these
quantities.
Some irreducible tensorial sets.
Beside the standard irreducible tensorial sets such as vector, scalers, spherical harmonics,
etc. that you are no doubt familiar with there are two other sets of irreducible tensorial sets
that are employed with great frequency in applications. One of the most imporatant sets
is the statistical tensors defined by Fano. Thus consider a density matrix element ρjm,j 0 m0
whose elements define the state of a system for which angular momentum is a good quantum
number. This set of quantities does not transform as an irreducible set, but we can form
linear combinations that do. They are denoted by ρkq and called the state multipoles or
statistical tensors. We define
ρkq =
X
0
ρjmj 0 m0 (j − mj 0 m0 |kq)(−1)k−j −m
(10)
mm0
We will also use the projection operator defined as
Pqk =
X
0
|j 0 m0 >< jm|(jmj 0 − m0 |kq)(−1)k−j −m
(11)
mm0
Note that when k = 0 and j = j 0 this operator is just a scaler
P = (2j + 1)−1/2
X
|jm0 >< jm|
(12)
mm0
We now have most of the elements of the algebra, that is we have some examples of
irreducible tensorial sets and the most important operations that will be performed with
them. These few properties will enable us to rewrite expressions of measured quantities so
that geometrical properties under control of the observer are explicitly factored out of the
expressions. Later on, one may want to actually compute some of the quantities that appear
as coefficients in the equations. For this purpose one more formula is needed, namely the
Wigner-Eckart theorm. This theorm states
√
0
k
< αjm|T−q
|α0 j 0 m0 >= (αj||T k ||α0 j 0 )(−1)−j+k−m (kq|j − mj 0 m0 )/ 2k + 1
(13)
[Note: This definition may also be written
√
< α0 j 0 m0 |Tqk |αjm > = (α0 j 0 ||T k ||αj)(−1)j−m (kq|j 0 m0 j − m)/ 2k + 1
√
0
= (α0 j 0 ||T k ||αj)(−1)−k−j −m (kq|j − mj 0 − m0 )/ 2k + 1
4
(14)
(15)
so that
< Tqk >≡
X
ρjmj 0 m0 < j 0 m0 |Tqk |jm >= ρkq
mm0
< j 0 ||T k ||j >
√
(−1)2k
2k + 1
(16)
] This expression complements our statements concern the coupling of matrix elements in
that it factors out a purely geometrical factor, the Clebsh-Gordan coefficient, from the
dynamical factor represented by the reduced matrix element
(αj||T k ||α0 j 0 )
.
Note on allowed tensorial ranks.
Our notes have not specified the tensorial ranks that are employed in the Fano-Racah
tensorial sets. Integer ranks are natually generated by coupling vector sets to form sets of
any integer rank. As seen from Eq. (1) the corresponding transformation matrices are have
dimensions 2k + 1 which are odd integer only. On may well ask why there not even simpler
nontrivial representations in terms of two dimensional matrices. Indeed, one can find such
representations if one can find irreducible sets with only two components. This was done
by Cartan even before the physical content of such sets became apparent in the quantum
theory. We will now construct such sets following, roughly, the outline of Cartan’s approach.
Let us postulate an irreducible set with two componets. This set is given the rank k = 1/2
in order to preserve the relation that its number of components ( two) is equal to 2k + 1. In
order to fit this new set into our algebra it must be possible to form all possible products
1/2
1/2
T1/2 T1/2 ,
1/2
1/2
1/2
T1/2 T−1/2 ,
1/2
T−1/2 T1/2 ,
1/2
1/2
T−1/2 T−1/2
of this set with itself and obtain a scaler and a vector by the process of reduction . It is easy
1/2
1/2
1/2
1/2
to see that the combination T1/2 T−1/2 − T−1/2 T1/2 = 0 is a scaler since it is identically zero
and thus unchanged by coordinate transformations. It follows that linear combinations of the
other three components form the components of a vector. These are easily contructed and do
indeed transform under coordinate rotations as the components of a vector, but the vector
√
1/2 1/2
1/2
1/2
1/2
1/2
1/2 1/2
has zero length. [Note: v+ = T1/2 T1/2 , v= T−1/2 T−1/2 , v0 = (T−1/2 T1/2 +T1/2 T−1/2 )1/ 2 = 0
so that v 2 = −(v+ v− + v− v+ ) + v0 v0 = 0] This is of no concern if we are only intereseted is
constructing a possible set of rank 1/2, but the results are more transparent if we use two
sets of rank 1/2 to construct a conventional coordinate vector by the standard process of
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forming all possible products then reducing the set to its irreducible components which are
a scaler and the three components of a coordinate vector. Thus we seek sets T1/2 and S1/2
chosen so that the tensor [T1/2 ×S1/2 ]1 equals the coordinate vector r. The reader can verify
that the choice:
1/2
T−1/2 =
√
√
1
1
1/2
r cos θe−iφ/2 , T1/2 = r sin θe+iφ/2
2
2
(17)
with
1/2
Sq1/2 = (−1)1/2+q (T−q )∗
together with appropriate Clebsch-Gordan coefficients ( 21 q 12 q 0 |1m) do indeed yield the coordinate vector. Similarly, the scaler [T1/2 ×S1/2 ]0 is proportional to the length of the coordinate
vector r. This verifies that we can indeed include sets of rank 1/2 in our algebra. Furthermore the sets of rank 1/2 can, by the process of reduction, be used to generate sets of any
rank.
Since observable quantities namely the elements of the density matrix are emphasized
in these notes, and since they are formed from state vectors by taking matrix elements,
they always involve two irreducible sets. Thus the irreducible sets of observables, that is
the state multipoles, are sets of integer rank only, just as in classical mechanics. Thus sets
of rank 1/2 are allowed in both classical and quantum mechanics, but never play any role
in classical physics. In classical mechanics an elementary particle may have a magnetic
moment without having a quadrupole moment, whereas in quantum mechanics this is not
easily accomplished if only sets of integer rank are allowed. Alternatively, an elementary
particle with spin 1/2 can have only a magnetic moment without a quadrupole moment since
the observable statistical multipoles for this system are of rank 0 and 1. The existance of
systems which exhibit only a magnetic moment without a corresponding quadrupole moment
indicates that sets with rank 1/2 are relevant to the description of nature, thus it is essential
to include them in our algebra.
Application to photoionization
Consider photoionisation from a state charactarized by a density matrix element ρmi ,m0i .
We have for the photoabsorption A coefficient the expression
A=K
X
mi ,m0i
< ψf |εg · r|imi > ρmi ,m0i < im0i |ε∗g · r0 |ψf >,
6
(18)
where K is a constant whose value does not concern us here. This expression is then written
in terms of state mulipoles ρkq and the projection operator Pqk of Eq. (12). Using the FanoRacah notation introduced in Eq. (4) we have
A = K3
X
0
(2k + 1)1/2 < ψf |[εg × r]0 [ε∗g × r0 ] [Pk (r, r0 )×ρk ]0 |ψf >
(19)
k
We now recouple the irreducible tensors to group all factors depending upon coordinates
k
together to form a tensor < ψf |[[r × r0 ]k1 × Pk (r, r0 )] 2 |ψf >. This require two recouplings
so that we have
A = K3
X
(2k + 1)1/2 ((11)0, (11)0, 0|(11)k1 , (11)k1 , 0)
k,k1 ,k2
×((k1 k1 )0, (k1 k1 )0, 0|(k1 k)k2 , (k1 k)k2 , 0)
h
k
i0
× [[εg × ε∗g ]k1 ×ρk ]k2 < ψf |[[r × r0 ] 1 ×Pk (r, r0 )]k2 |ψf >
(20)
We now need to extract the geometrical factors from the dynamical matrix element. Thus
we consider the matrix element in a frame with the wave vector of the detected electron ke
parallel to the z-axis. Then the matrix element
k
< ψf |[[r × r0 ] 1 ×Pk (r, r0 )]kq 2 |ψf >
(21)
vanishes unless q = 0. Denote this matrix element which only depends upon the direction
of ke by the symbol M k2 (ke ). We then rotate to an arbitary frame to obtain the matrix
element in such a frame. Let the anglular coordinates of ke in this frame be θe , φe = k̂e then
we have
k
k2
< ψf |[[r × r0 ] 1 ×Pk (r, r0 )]kq 2 |ψf >= Dq,0
(φe , θe , 0)M k2
k2
Dq,0
(φe , θe , 0)
4π
=
2k2 + 1
1/2
Yqk2 (k̂e )
(22)
(23)
Upon subsituting Eq. ( 16) into Eq. (13 ) we obtain an expression for A which has all of the
geometrical factors explicitly exhibited
√ X
A = K6 π
!
k,k1 ,k2
2k + 1
((11)0, (11)0, 0|(11)k1 , (11)k1 , 0)
2k2 + 1
×((k1 k1 )0, (k1 k1 )0, 0|(k1 k)k2 , (k1 k)k2 , 0)
h
k1
i0
k2
× [[εg × ε∗g ] ×ρk ] ×Yk2 M k2 (ke )
7
(24)
Note that we have extracted all of the geometrical factors without introducing a partial wave
expansion. That is only needed to compute the set of parameter M k2 (ke ) using a particular
dynamical theory, a completely separate task from extracting the geometrical factors of a
measurement. Here we regard the quantities M k2 (ke ) as parameters to fit to experimental
measurements.
To complete our task we now consider the state multipoles for a particular experiment.
We suppose that the state multipoles pertain to a state prepared by exciting systems-atoms
or molecules-initially in isotropic states by optical excitation. Consider that the dipole
approximation holds and that there is no optical pumping. Then the population of state i is
determined by a dipole transition and we may write the unormalized density matrix element
as
ρmi ,m0i =
X
< imi |εb · r|omo >< omo |εb · r|im0i >
(25)
m0
This sum has much the same form as Eq.(15) so that we can apply exactly the same
recoupling transformations as before to obtain
ρmi ,m0i =
X
| < imi |εb · r|omo >< omo |εb · r|im0i >
m0
0
= 3(2jo + 1)1/2 < imi |[[εb × r]0 × [ε∗b × r0 ]0 ] P 0 (r, r0 )|im0i >
= 3(2jo + 1)1/2
X
((11)0, (11)0, 0|(11)k3 , (11)k3 , 0)
k3
×[[εb × ε∗b ]k3 < imi |[r × r0 ]k3 P 0 (r, r0 )|im0i > ]0
(26)
We now employ the Wigner-Eckardt theorem to write
ρmi ,m0i = 3(2jo + 1)1/2
X
((11)0, (11)0, 0|(11)k3 , (11)k3 , 0)
k3
k
×[εb × ε∗b ]Ik3q3 (2k3 + 1)−1/2 < imi |[r × r0 ]−q3 3 P0 (r, r0 )|im0i >
= 3(2jo + 1)1/2
X
((11)0, (11)0, 0|(11)k3 , (11)k3 , 0)[εb × ε∗b ]kq33
k3
0
0 k3
< i||[r × r ] P0 (r, r0 )||i0 > (−1)ji +k3 −mi (k3 q3 |ji − mi ji0 m0i )
(27)
This expression is now substituted into Eq.(10) to obtain the state multipoles. Using the
orthognality properties of the Clebsh-Gordan coefficients we obtain
ρkq = 3(2jo + 1)1/2 ((11)0, (11)0, 0|(11)k, (11)k, 0)
k
0
0
×[εb × ε∗b ]kq < i||[r × r0 ] P 0 (r, r0 )||i0 > (−1)−ji −mi −ji −mi
≡ N k [εb × ε∗b ]kq
(28)
8
Here the parameter N k is regarded as a dynamical quantity whose calculation represents a
separate task. The final form of our expression for the quantitity A in volves a sum over
terms of the form
h
k1
[[εg × ε∗g ] ×[εb × ε∗b ]k ]k2 ×Yk2
i0
(29)
Our final result is
√ X
A = K6 π
!
k,k1 ,k2
2k + 1
((11)0, (11)0, 0|(11)k1 , (11)k1 , 0)
2k2 + 1
×((k1 k1 )0, (k1 k1 )0, 0|(k1 k)k2 , (k1 k)k2 , 0)M k2 N ke
k1
h
× [[εg × ε∗g ] ×[εb × ε∗b ]k ]k2 ×Yk2
i0
(30)
Final remark.
In this derivation we have omitted the effects of hyperfine structure of the state i and
finite laser pulse duration on the state multipole. We easily include these effects by the
substitution
k
N →N
k
Z
dtdt0 fb (t)fg (t0 )Gk (t − t0 )
(31)
where fb (t) and fg (t) are functions that describe the time dependent laser intensity, and
Gk (t − t0 ) is the perturbation coefficient given by Fano and Macek, for example.
The above expression expresses the dependence of A upon the geometrical parameters of
the experimental apparatus. It was obtained using only a minimum of dynamical input, that
is, we employed only the dipole approximation and conservation of angular momentum [using
the Wigner-Eckardt theorem]. The dynamical aspects are incorporated in the parameters
M k2 (ke ) and N k . Computation of these quantities on the basis of a specific theory is a
separate task. As the parameter N k depends only upon the reduced dipole matrix elements
< i||r||o > it can accordingly be further simplified. These are tasks that require us to
use formula of angular momentum theory that go beyond the manipulation of irreducible
tensors. It will be necessary to introduce properties of state vectors instead of just the
properties of the projection operator constructed from state vectors.
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