Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
Подождите немного. Документ загружается.
68 Part A Mathematical Methods
Using the definitions introduced above, we can now
give the generating functions for the coupled wave func-
tions and the recoupling coefficients for each coupling
scheme as determined by the bracketing B.
Generating function for coupled wave functions:
e
xA(T)y
T
= e
8
1≤i< j≤n+1
a
i, j
det(z
i
,z
j
)
=
jk
Φ
B
j,k, j
(T )
m
(−1)
j−m
ψ
j,−m
(x
n+1
, y
n+1
)
× Ψ
B
( jk) jm
(x, y), (2.142)
x = (x
1
, x
2
,... ,x
n+1
), y = (y
1
, y
2
,... ,y
n+1
).
Relation to U(n+1) solid harmonics:
e
xA(T)y
T
=
∞
k=0
α,βk
x
α
√
α!
D
k
α,β
(A(T))
y
β
√
β!
.
(2.143)
Relation of U(n +1) solid harmonics to triangle mono-
mials:
D
k
α,β
[A(T)]=(−1)
j−m
k
jkj
mqm
B
Φ
B
j,k, j
(T )
(2.144)
α
i
= j
i
+m
i
,β
i
= j
i
−m
i
i = 1, 2,... ,n ;
α
n+1
= j −m ,β
n+1
= j +m ;
i
α
i
=
i
β
i
= j
1
+ j
2
+···+j
n
+ j .
The relation between the skew symmetric matrix A(T)
of order n +1 and the elements of the 3 × (n −1) ma-
trix T is that described in relations (2.138).
Generating function for all recoupling coefficients:
1
[Pf(A(T), A
(T
))]
2
(2.145)
=
j,k,k
, j
Φ
B
j,k, j
(T )Φ
B
j
,k
, j
(T
)j, k, j||j
, k
, j,
where j, k, j || j
, k
, j denotes the recoupling co-
efficient that effects the transformation between the
coupling schemes corresponding to the bracketing B
and the bracketing B
, and where the sequence j
is
a permutation of j in accordance with the bracketing B
.
We also note that
1
Pf(A, A
)
=
1
√
det(I − AA
)
=
1 +
k≥1
α,βk
D
k
α,β
(A)D
k
β,α
(A
)
1
2
, (2.146)
for arbitrary skew symmetric matrices of order n.
Relation (2.145) generates all recoupling coeffi-
cients, the trivial ones (those differing by signs) and
all the complicated ones, that is, those corresponding
to 3n − j coefficients. It will also be observed that the
expansion of the reciprocal of the double Pfaffian ef-
fects an infinite sum in which no radicals occur, which
in turn implies that the every recoupling coefficient has
the form
j, k, j || j
, k
, j
=
a
i
,b
i
,k
i
∈T
B
( j,k, j)
{a
i
, b
i
, k
i
}
×
a
i
,b
i
,k
i
∈T
B
( j
,k
, j)
(
a
i
, b
i
, k
i
)
× I( j, k, j || j
, k
, j),
where I( j, k, j || j
, k
, j) is an integer:
Each recoupling coefficient is an integer multiplied by
square-root factors that depend on the triangles associ-
ated with the coupling scheme.
Such features can be very useful in the development
of algorithms for the calculation of 3n − j coefficients,
including WCG-coefficients [2.17]. Relation (2.145)
should be useful for the classification of 3n − j coef-
ficients.
Part A 2.13
Angular Momentum Theory 2.14 Tables 69
2.14 Tables
Excerpts and Fig. 2.1 are reprinted from Bieden-
harn and Louck [2.1] with permission of Cambridge
University Press. Tables 2.2–2.4 have been adapted
Table 2.1 The solid and spherical harmonics Y
lm
, and the tensor harmonics T
k
µ
(labeled by k =l and µ =m)forl = 0,
1, 2, 3, and 4
l m
√
4π
Y
lm
(r)
√
4π
Y
lm
(θ,ϕ) T
l
m
1 ±1 ∓
9
3
2
(x ±iy) ∓
9
3
2
e
±iϕ
sin θ ∓
√
2J
±
0
√
3 z
√
3cosθ 2J
3
2 ±2
1
2
9
15
2
(x ±iy)
2
1
2
9
15
2
e
±2iϕ
sin
2
θ
√
6
J
2
±
±1 ∓
9
15
2
(x ±iy)z ∓
9
15
2
e
±iϕ
sin θ cos θ ∓
√
6J
±
(2J
3
±1)
0
1
2
√
5 (3z
2
−r
2
)
1
2
√
5 (3cos
2
θ −1) 2
3J
2
3
− J
2
3 ±3 ∓
1
4
√
35 (x ±iy)
3
∓
1
4
√
35 e
±3iϕ
sin
3
θ ∓2
√
5J
3
±
±2
1
2
9
105
2
(x ±iy)
2
z
1
2
9
105
2
e
±2iϕ
sin
2
θ cos θ 2
√
30J
2
±
(J
3
±1)
±1 ∓
1
4
21(x ±iy)(5z
2
−r
2
) ∓
1
4
√
21 e
±iϕ
sin θ(5cos
2
θ −1) ∓2
√
3J
±
5J
2
3
− J
2
±5J
3
+2
0
1
2
√
7 (5z
2
−3r
2
)z
1
2
√
7 (5cos
2
θ −3) cos θ 4
5J
2
3
−3J
2
+1
J
3
4 ±4
3
16
√
70 (x ±iy)
4
3
16
√
70 e
±4iϕ
sin
4
θ
√
70J
4
±
±3 ∓
3
4
√
35 (x ±iy)
3
z ∓
3
4
√
35 e
±3iϕ
sin
3
θ cos θ ∓2
√
35J
3
±
(2J
3
±3)
±2
3
8
√
10 (x ±iy)
2
(7z
2
−r
2
)
3
8
√
10 e
±2iϕ
sin
2
θ(7cos
2
θ −1) 2
√
10J
2
±
7J
2
3
− J
2
±14J
3
+9
±1 ∓
3
4
√
5 (x ±iy)(7z
2
−3r
2
)z ∓
3
4
√
5e
±iϕ
sin θ(7cos
2
θ −3) cos θ ∓
√
5J
±
28J
3
3
−12J
2
J
3
±42J
2
3
−6J
2
+38J
3
±12
0
15
8
7z
4
−6z
2
r
2
+
3
5
r
4
15
8
7cos
4
θ −6cos
2
θ +
3
5
70J
4
3
−60J
2
J
2
3
+6(J
2
)
2
+50J
2
3
−12J
2
Table 2.2 The 3– j coefficients for all M’s = 0, or J
3
= 0,
1
2
J
1
J
2
J
3
000
=(−1)
1
2
J
(J
1
+ J
2
− J
3
)!(J
1
+ J
3
− J
2
)!(J
2
+ J
3
− J
1
)!
(J
1
+ J
2
+ J
3
+1)!
1
2
×
(
1
2
J)!
1
2
J − J
1
!
1
2
J − J
2
!
1
2
J − J
3
!
, J even
J
1
J
2
J
3
000
=0, J odd, where J = J
1
+ J
2
+ J
3
JJ0
M −M 0
=(−1)
J−M
1
(2J +1)
1/2
J +
1
2
J
1
2
M −M −
1
2
1
2
=(−1)
J−M−
1
2
J − M +
1
2
(2J +2)(2J +1)
1/2
from Edmonds [2.18] by permission of Prince-
ton University Press. Thanks are given for this
cooperation.
Part A 2.14
70 Part A Mathematical Methods
Table 2.3 The 3– j coefficients for J
3
= 1,
3
2
, 2
J +1 J 1
M −M −11
=(−1)
J−M−1
(J − M)(J −M +1)
(2J +3)(2J +2)(2 J +1)
1
2
J +1 J 1
M −M 0
=(−1)
J−M−1
2(J + M +1)(J − M +1)
(2J +3)(2J +2)(2 J +1)
1
2
JJ1
M −M−11
=(−1)
J−M
2(J − M)(J + M +1)
(2J +2)(2J +1)(2 J)
1
2
JJ1
M −M 0
=(−1)
J−M
M
[
(2J +1)(J +1) J
]
1
2
J +
3
2
J
3
2
M −M −
3
2
3
2
=(−1)
J−M+
1
2
J − M −
1
2
J − M +
1
2
J − M +
3
2
(2J +4)(2J +3)(2 J +2)(2J +1)
1
2
J +
3
2
J
3
2
M −M −
1
2
1
2
=(−1)
J−M+
1
2
3
J − M +
1
2
J − M +
3
2
J + M +
3
2
(2J +4)(2J +3)(2 J +2)(2J +1)
1
2
J +
1
2
J
3
2
M −M −
3
2
3
2
=(−1)
J−M−
1
2
3
J − M −
1
2
J − M +
1
2
J + M +
3
2
(2J +3)(2J +2)(2 J +1)2J
1
2
J +
1
2
J
3
2
M −M −
1
2
1
2
=(−1)
J−M−
1
2
J − M +
1
2
(2J +3)(2J +2)(2 J +1)2J
1
2
J +3M +
3
2
J +2 J 2
M −M −22
=(−1)
J−M
(J − M −1)(J − M)(J −M +1)(J −M +2)
(2J +5)(2J +4)(2 J +3)(2J +2)(2J +1)
1
2
J +2 J 2
M −M −11
=(−1)
J−M
(J + M +2)(J − M +2)(J −M +1)(J − M)
(2J +5)(2J +4)(2 J +3)(2J +2)(2J +1)
1
2
J +2 J 2
M −M 0
=(−1)
J−M
6(J + M +2)(J + M +1)(J −M +2)(J − M +1)
(2J +5)(2J +4)(2 J +3)(2J +2)(2J +1)
1
2
J +1 J 2
M −M −22
=2(−1)
J−M+1
(J − M −1)(J − M)(J −M +1)(J +M +2)
(2J +4)(2J +3)(2 J +2)(2J +1)2J
1
2
J +1 J 2
M −M −11
=(−1)
J−M+1
2(J +2M +2)
(J − M +1)(J −M)
(2J +4)(2J +3)(2 J +2)(2J +1)2J
1
2
J +1 J 2
M −M 0
=(−1)
J−M+1
2M
6(J + M +1)(J − M +1)
(2J +4)(2J +3)(2 J +2)(2J +1)2J
1
2
JJ2
M −M−22
=(−1)
J−M
6(J − M −1)(J − M)(J +M +1)(J + M +2)
(2J +3)(2J +2)(2 J +1)(2J)(2J −1)
1
2
JJ2
M −M−11
=(−1)
J−M
(1 +2M)
6(J + M +1)(J − M)
(2J +3)(2J +2)(2 J +1)(2J)(2J −1)
1
2
JJ2
M −M 0
=(−1)
J−M
2
!
3M
2
− J(J +1)
"
[
(2J +3)(2J +2)(2 J +1)(2J)(2J −1)
]
1
2
Part A 2.14
Angular Momentum Theory 2.14 Tables 71
Table 2.4 The 6– j coefficients for d =0,
1
2
, 1,
3
2
, 2, with s =a+b+c
abc
0 ef
=(−1)
s
[
(2b+1)(2c+1)
]
−1
2
δ
bf
δ
ce
ab c
1
2
c−
1
2
b+
1
2
=(−1)
s
(s−2b)(s−2c+1)
(2b+1)(2b+2)2c(2c+1)
1
2
ab c
1
2
c−
1
2
b−
1
2
=(−1)
s
(s+1)(s−2a)
2b(2b+1)2c(2c+1)
1
2
ab c
1 c−1 b−1
= (−1)
s
s(s+1)(s−2a−1)(s−2a)
(2b−1)2b(2b+1)(2c−1)2c(2c+1)
1
2
abc
1 c−1 b
=(−1)
s
2(s+1)(s−2a)(s−2b)(s−2c+1)
2b(2b+1)(2b+2)(2c−1)2c(2c+1)
1
2
ab c
1 c−1 b+1
=(−1)
s
(s−2b−1)(s −2b)(s −2c+1)(s−2c+2)
(2b+1)(2b+2)(2b+3)(2c−1)2c(2c+1)
1
2
abc
1 cb
=(−1)
s+1
2
[
b(b+1) +c(c+1) −a(a+1)
]
[
2b(2b+1)(2b+2)2c(2c+1)(2c+2)
]
1
2
,
ab c
3
2
c−
3
2
b−
3
2
=(−1)
s
(s−1)s(s+1)(s−2a−2)(s−2a−1)(s−2a)
(2b−2)(2b−1)2b(2b+1)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
3
2
c−
3
2
b−
1
2
=(−1)
s
3s(s+1)(s−2a−1)(s−2a)(s−2b)(s−2b+1)
(2b−1)2b(2b+1)(2b+2)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
3
2
c−
3
2
b+
1
2
=(−1)
s
3(s+1)(s−2a)(s−2b−1)(s −2b)(s−2c+1)(s−2c+2)
2b(2b+1)(2b+2)(2b+3)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
3
2
c−
3
2
b+
3
2
=(−1)
s
(s−2b−2)(s −2b−1)(s−2b)(s−2c+1)(s −2c+2)(s−2c+3)
(2b+1)(2b+2)(2b+3)(2b+4)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
3
2
c−
1
2
b−
1
2
=(−1)
s
[
2(s−2b)(s−2c) −(s +2)(s −2a−1)
][
(s+1)(s−2a)
]
1
2
[
(2b−1)2b(2b+1)(2b+2)(2c−1)2c(2c+1)(2c+2)
]
1
2
ab c
3
2
c−
1
2
b+
1
2
=(−1)
s
[
(s−2b−1)(s −2c) −2(s+2)(s−2a)
][
(s−2b)(s−2c+1)
]
1
2
[
2b(2b+1)(2b+2)(2b+3)2c(2c+1)(2c+2)(2c+3)
]
1
2
,
ab c
2 c−2 b−2
=(−1)
s
(s−2)(s−1)s(s+1)
(2b−3)(2b−2)(2b−1)2b(2b+1)
1
2
×
(s−2a−3)(s−2a−2)(s −2a−1)(s −2a)
(2c−3)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
2 c−2 b−1
=(−1)
s
2
(s−1)s(s+1)
(2b−2)(2b−1)2b(2b+1)(2b+2)
1
2
×
(s−2a−2)(s−2a−1)(s −2a)(s−2b)(s−2c+1)
(2c−3)(2c−2)(2c−1)2c(2c+1)
1
2
Part A 2.14
72 Part A Mathematical Methods
Table 2.4 The 6– j coefficients for d = 0,
1
2
, 1,
3
2
, 2, with s =a+b+c, cont.
abc
2 c−2 b
= (−1)
s
6s(s+1)(s−2a−1)(s−2a)
(2b−1)2b(2b+1)(2b+2)(2b+3)
1
2
×
(s−2b)(s−2c+1)(s −2c+2)
(2c−3)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
2 c−2 b+1
= (−1)
s
2
(s+1)(s−2a)(s−2b−2)(s−2b−1)(s−2b)
2b(2b+1)(2b+2)(2b+3)(2b+4)
1
2
×
(s−2c+1)(s −2c+2)(s−2c+3)
(2c−3)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
2 c−2 b+2
= (−1)
s
(s−2b−3)(s −2b−2)(s−2b−1)(s −2b)
(2b+1)(2b+2)(2b+3)(2b+4)(2b+5)
1/2
×
(s−2c+1)(s −2c+2)(s−2c+3)(s −2c+4)
(2c−3)(2c−2)(2c−1)2c(2c+1)
1
2
ab c
2 c−1 b−1
= (−1)
s
4
[
(a+b)(a−b+1) −(c−1)(c−b+1)
]
[
(2b−2)(2b−1)2b(2b+1)(2b+2)
]
1
2
×
s(s+1)(s−2a−1)(s−2a)
(2c−2)(2c−1)2c(2c+1)(2c+2)
1
2
abc
2 c−1 b
= (−1)
s
2
!
(a−b+1)(a−b) −c
2
+1
"
[
(2b−1)2b(2b+1)(2b+2)(2b+3)
]
1
2
×
6(s+1)(s−2a)(s−2b)(s−2c+1)
(2c−2)(2c−1)2c(2c+1)(2c+2)
1
2
ab c
2 c−1 b+1
= (−1)
s
4
[
(a+b+2)(a−b−1) −(c−1)(b+c+2)
]
[
2b(2b+1)(2b+2)(2b+3)(2b+4)
]
1
2
×
(s−2b−1)(s −2b)(s −2c+1)(s−2c+2)
(2c−2)(2c−1)2c(2c+1)(2c+2)
1
2
,
abc
2 cb
= (−1)
s
2
[
3X(X −1) −4b(b+1)c(c+1)
]
[
(2b−1)2b(2b+1)(2b+2)(2b+3)
]
1
2
×
1
(2c−1)2c(2c+1)(2c+2)(2c+3)
1
2
,
where X =b(b+1) +c(c+1) −a(a+1)
References
2.1 L. C. Biedenharn, J. D. Louck: Encyclopedia of Math-
ematics and Its Applications,Vol.8&9,ed.by
G.-C. Rota (Addison-Wesley, Reading 1981) presently
by (Cambridge Univ. Press, Cambridge)
2.2 J. D. Louck: J. Math. and Math. Sci. 22, 745 (1999)
2.3 J. Schwinger: On Angular Momentum. U. S. Atomic
Energy Commission Report NYO-3071, 1952 (unpub-
lished). In: Quantum Theory of Angular Momentum,
ed. by L. C. Biedenharn, H. van Dam (Academic, New
York 1965) pp. 229–279
2.4 B. R. Judd, G. M. S. Lister: J. Phys. A 20, 3159 (1987)
2.5 B. R. Judd: Symmetries in Science, ed. by B. Gruber,
R. S. Millman (Plenum, New York 1980) pp. 151–160
2.6 S. Roman, G.-C. Rota: Adv. in Math. 27, 95 (1978)
2.7 A. P. Jucys, I. B. Levinson, V. V. Vanagas: The Theory
of Angular Momentum (Israel Program for Scien-
tific Translation, Jerusalem 1962) (Mathematicheskii
apparat teorii momenta kolichestva dvizheniya)
Translated from the Russian by A. Sen, A. R. Sen
(1962)
Part A 2
Angular Momentum Theory References 73
2.8 A. P. Jucys, A. A. Bandzaitis: Angular Momentum
Theory in Quantum Physics (Moksias, Vilnius 1977)
2.9 R. P. Stanley: Enumerative Combinatorics,Vol.1
(Cambridge Univ. Press, Cambridge 1997)
2.10 A. Clebsch: Theorie der binären algebraischen For-
men (Teubner, Leipzig 1872)
2.11 J. D. Louck, W. Y. C. Chen, H. W. Galbraith: Symme-
try, Structural Properties of Condensed Matter,ed.by
T. Lulek, B. Lulek, A. Wal (World Scientific, Singapore
1999) pp. 112–137
2.12 W. Y. C. Chen, J. D. Louck: Adv. Math. 140, 207 (1998)
2.13 P. A. MacMahon: Combinatory Analysis (Cambridge
Univ. Press, Cambridge 1960) (Chelsia Publishing Co.,
New York, 1960) (Originally published in two volumes
by Cambridge Univ. Press, Cambridge, 1915, 1916)
2.14 J. D. Louck: Adv. Appl. Math. 17, 143 (1996)
2.15 T. Molien: Über die Invarianten der linearen Substi-
tutionsgruppen, Sitzungsber. Konig. Preuss. Akad.
Wiss. 52, 1152 (1897)
2.16 L. Michel, B. I. Zhilinskii: Physics Reports 341, 11 (2001)
2.17 L. Wei: Comput. Phys. Commun. 120, 222 (1999)
2.18 A. R. Edmonds: Angular Momentum in Quantum Me-
chanics (Princeton Univ. Press, Princeton 1957)
2.19 E. Cartan: Thesis (Paris, Nony 1894) [Ouevres Com-
plète, Part 1, pp. 137–287 (Gauthier-Villars, Paris
1952)]
2.20 H. Weyl: Gruppentheorie und Quantenmechanik
(Hirzel, Leipzig 1928) Translated by H. P. Robertson
as The Theory of Groups and Quantum Mechanics
(Methuen, London 1931)
2.21 P. A. M. Dirac: The Principles of Quantum Mechanics,
4th edn. (Oxford Univ. Press, London 1958)
2.22 M. Born, P. Jordan: Elementare Quantenmechanik
(Springer, Berlin, Heidelberg 1930)
2.23 H. B. G. Casimir: Thesis, University of Leyden (Wolters,
Groningen 1931) [Koninkl. Ned. Akad. Wetenschap,
Proc. 34, 844 (1931)]
2.24 B. L. van der Waerden: Die gruppentheoretische
Methode in der Quantenmechanik (Springer, Berlin,
Heidelberg 1932)
2.25 W. Pauli: Handbuch der Physik,Vol.24,ed.by
H. Geiger, K. Scheel (Springer, Berlin, Heidelberg
1933) Chap. 1, pp. 83–272. Later published in Ency-
clopedia of Physics, Vol. 5, Part 1, ed. by S. Flügge
(Springer, Berlin, Heidelberg 1958), pp. 45, 46
2.26 H. Weyl: The Structure and Representations of
Continuous Groups, Lectures at the Institute for Ad-
vanced Study. Princeton, 1934-1935 (unpublished).
Notes by R. Brauer
2.27 E. U. Condon, G. H. Shortley: The Theory of Atomic
Spectra (Cambridge Univ. Press, London 1935)
2.28 H. A. Kramers: Quantum Mechanics (North-Holland,
Amsterdam 1957) Translation by D. ter Haar of
Kramer’s monograph published in the Hand- und
Jahrbuch der chemischen Physik (1937)
2.29 G. Szegö: Orthogonal Polynomials (Edwards, Ann
Arbor 1948)
2.30 I. M. Gel’fand, Z. Ya. Shapiro: Am. Math. Soc. Transl.
2, 207 (1956)
2.31 A. Erdelyi, W. Magnus, F. Oberhettinger, G. F. Tri-
comi: Higher Transcendental Functions,Vol.1
(McGraw-Hill, New York 1953)
2.32 E. P. Wigner: Application of Group Theory to the
Special Functions of Mathematical Physics. Lecture
notes, 1955 (unpublished)
2.33 H. C. Brinkman: Applications of Spinor Invariants in
Atomic Physics (North-Holland, Amsterdam 1956)
2.34 M. E. Rose: Elementary Theory of Angular Momentum
(Wiley, New York 1957)
2.35 U.Fano,G.Racah:Irreducible Tensorial Sets (Aca-
demic, New York 1959)
2.36 E. P. Wigner: Group Theory and Its Application to the
Quantum Mechanics of Atomic Spectra (Academic,
New York 1959) Translation from the 1931 German
edition by J. J. Griffin
2.37 J. C. Slater: Quantum Theory of Atomic Structure,
Vol. 2 (McGraw-Hill, New York 1960)
2.38 V. Heine: Group Theory and Quantum Mechanics; An
Introduction to Its Present Usage (Pergamon, New
York 1960)
2.39 W. T. Sharp: Thesis, Princeton University (1960) (is-
sued as Report AECL-1098, Atomic Energy of Canada,
Chalk River, Ontario (1960))
2.40 D. M. Brink, G. R. Satchler: Angular Momentum (Ox-
ford Univ. Press, London 1962)
2.41 M. Hamermesh: Group Theory and Its Applications to
Physical Problems (Addison-Wesley, Reading 1962)
2.42 G. W. Mackey: The Mathematical Foundations
of Quantum Mechanics (Benjamin, New York
1963)
2.43 A. de-Shalit, I. Talmi: Nuclear Shell Theory (Pure and
Applied Physics Series), Vol. 14 (Academic, New York
1963)
2.44 R. P. Feynman: Feynman Lectures on Physics
(Addison-Wesley, Reading 1963) Chap. 34
2.45 R. Judd: Operator Techniques in Atomic Spectroscopy
(McGraw-Hill, New York 1963)
2.46 A. S. Davydov: Quantum Mechanics (Pergamon,
London, Addison-Wesley, Reading 1965) Transla-
tion from the Russian of Kvantovaya Mekhanika
(Moscow, 1963), with revisions and additions by
D. ter Haar
2.47 I. M. Gel’fand, R. A. Minlos, Z. Ya. Shapiro: Represen-
tations of the Rotation and Lorentz Groups and Their
Applications (Macmillan, New York 1963) Translated
from the Russian by G. Cummins and T. Boddington
2.48 R. Hagedorn: Selected Topics on Scattering The-
ory: Part IV, Angular Momentum, Lectures given
at the Max-Planck-Institut für Physik, Munich
(1963)
2.49 M. A. Naimark: Linear Representations of the Lorentz
Group (Pergamon, New York 1964)
2.50 L. C. Biedenharn, H. van Dam: Quantum Theory of
Angular Momentum (Academic, New York 1965)
Part A 2
74 Part A Mathematical Methods
2.51 B. L. van der Waerden: Sources of Quantum Mechan-
ics (North-Holland, Amsterdam 1967)
2.52 B. R. Judd: Second Quantization and Atomic Spec-
troscopy (Johns Hopkins Press, Baltimore 1967)
2.53 N. Vilenkin: Special Functions and the Theory of
Group Representations, Vol. 22, Translated from the
Russian Am. Math. Soc. Transl. (Amer. Math. Soc.,
Providence, 1968)
2.54 J. D. Talman: Special Functions: A Group Theoretic
Approach (Benjamin, New York 1968) Based on E. P.
Wigner’s lectures (see [2.3])
2.55 B. G. Wybourne: Symmetry Principles and Atomic
Spectroscopy (Wiley-Interscience, New York 1970)
2.56 E. A. El Baz, B. Castel: Graphical Methods of Spin
Algebras in Atomic, Nuclear, and Particle Physics
(Dekker, New York 1972)
2.57 R. Gilmore: Lie Groups, Lie Algebras, and Some of
Their Applications (Wiley, New York 1974)
2.58 D. A. Varshalovich, A. N. Moskalev, V. K. Kherson-
ski
ˇ
ı: Quantum Theory of Angular Momentum (Nauka,
Leningrad 1975) (in Russian)
2.59 R. D. Cowan: The Theory of Atomic Structure and
Spectra (Univ. Calif. Press, Berkeley 1981)
2.60 R. N. Zare: Angular Momentum (Wiley-Interscience,
New York 1988)
2.61 G. E. Andrews, R. A. Askey, R. Roy: Special Functions.
In: Encyclopidia of Mathematics and Its Applica-
tions, Vol. 71, ed. by G.-C. Rota (Cambridge Univ.
Press, Cambridge 1999)
2.62 P. Gordan:
¨
Uber das Formensystem binärer Formen
(Teubner, Leipzig 1875)
2.63 W. Heisenberg: Z. Phys. 33,879(1925)
2.64 M. Born, P. Jordan: Z. Phys. 34, 858 (1925)
2.65 P. A. M. Dirac: Proc. Soc. A 109,642(1925)
2.66 M. Born, W. Heisenberg, P. Jordan: Z. Phys. 35, 557
(1926)
2.67 W. Pauli: Z. Phys. 36, 336 (1926)
2.68 E. P. Wigner: Z. Phys. 43,624(1927)
2.69 E. P. Wigner: Z. Phys. 45, 601 (1927)
2.70 C. Eckart: Rev. Mod. Phys. 2, 305 (1930)
2.71 E. P. Wigner: Göttinger Nachr., Math.-Phys. 546,
(1932)
2.72 J. H. Van Vleck: Phys. Rev. 47,487(1935)
2.73 E. P. Wigner: On the Matrices which Reduce the Kro-
necker Products of Representations of S. R. Groups,
1940 (unpublished). In: Quantum Theory of Angu-
lar Momentum, ed. by L. C. Biedenharn, H. van Dam
(Academic, New York 1965) pp. 87–133
2.74 G. Racah: Phys. Rev. 62,438(1942)
2.75 G. Racah: Phys. Rev. 63,367(1943)
2.76 I. M. Gel’fand, M. L. Tseitlin: Dokl. Akad. Nauk SSSR
71, 825 (1950)
2.77 J. H. Van Vleck: Rev. Mod. Phys. 23, 213 (1951)
2.78 H. A. Jahn: Proc. R. Soc. A 205, 192 (1951)
2.79 L. C. Biedenharn, J. M. Blatt, M. E. Rose: Rev. Mod.
Phys. 24,249(1952)
2.80 L. C. Biedenharn: Notes on Multipole Fields,Lecture
notes at Yale University, New Haven 1952 (unpub-
lished)
2.81 J. P. Elliott: Proc. R. Soc. A 218,345(1953)
2.82 L. C. Biedenharn: J. Math. Phys. 31, 287 (1953)
2.83 H. A. Jahn, J. Hope: Phys. Rev. 93,318(1954)
2.84 T. Regge: Nuovo Cimento 10,544(1958)
2.85 T. Regge: Nuovo Cimento 11, 116 (1959)
2.86 V. Bargmann: Commun. Pure Appl. Math. 14,187
(1961)
2.87 A. Giovannini, D. A. Smith: Spectroscopic and Group
Theoretic Methods in Physics (Racah Memorial Vol-
ume), ed. by F. Block, S. G. Cohen, A. de-Shalit,
S. Sambursky, I. Talmi (Wiley–Interscience, New York
1968) pp. 89–97
2.88 V. Bargmann: Rev. Mod. Phys. 34, 829 (1962)
2.89 L. Michel: Lecture Notes in Physics: Group Rep-
resentations in Mathematics and Physics, Battelle
Recontres, ed. by V. Bargmann (Springer, Berlin,
Heidelberg 1970) pp. 36–143
2.90 A. C. T. Wu: J. Math. Phys. 13, 84 (1972)
2.91 Ya. A. Smorodinski
˘
ı, L. A. Shelepin: Sov. Phys. Usp.
15,1(1972)
2.92 Ya. A. Smorodinski
˘
ı, L. A. Shelepin: Usp. Fiz. Nauk
106,3(1972)
2.93 L. A. Shelepin: Invariant algebraic methods and
symmetric analysis of cooperative phenomena.
In: Group-Theoretical Methods in Physics,ed.by
D. V. Skobel’tsyn (Fourth Internat. Collog., Nijmegen
1975) pp. 1–109 A special research report translated
from the Russian by Consultants Bureau, New York,
London.
2.94 Ya. A. Smorodinski
ˇ
ı: Sov. Phys. JETP 48,403
(1978)
2.95 I. M. Gel’fand, M. I. Graev: Dokl. Math. 33, 336 (2000)
Tranl. from Dokl. Akad. Nauk 372, 151 (2000)
Part A 2
75
Group Theory
3. Group Theory for Atomic Shells
The basic elements of the theory of Lie groups
and their irreducible representations (IRs)
are described. The IRsareusedtolabelthe
states of an atomic shell and also the com-
ponents of operators of physical interest.
Applications of the generalized Wigner-Eckart
theorem lead to relations between matrix
elements appearing in different electronic
configurations. This is particularly useful in
the f shell, where transformations among
the seven orbital states of an f electron can
be described by the unitary group U(7) and
its sequential subgroups SO(7), G
2
, and SO(3)
with respective IRs[λ], W, U,andL. Exten-
sions to groups that involve electron spin S
(like Sp(14)) are described, as are groups that
do not conserve electron number. The most
useful of the latter is the quasispin group
whose generators Q connect states of iden-
tical W, U, L and seniority v in the f shell.
The symmetries of products of objects (states
or operators) that themselves possess sym-
metries are described by the technique of
plethysms.
3.1 Generators.......................................... 75
3.1.1 Group Elements ........................ 75
3.1.2 Conditions on the Structure
Constants ................................. 76
3.1.3 Cartan–Weyl Form ..................... 76
3.1.4 Atomic Operators as Generators .. 76
3.2 Classification of Lie Algebras................. 76
3.2.1 Introduction ............................. 76
3.2.2 The Semisimple Lie Algebras ....... 76
3.3 Irreducible Representations.................. 77
3.3.1 Labels ...................................... 77
3.3.2 Dimensions .............................. 77
3.3.3 Casimir’s Operator ..................... 77
3.4 Branching Rules .................................. 78
3.4.1 Introduction ............................. 78
3.4.2 U(n) ⊃ SU(n)............................. 78
3.4.3 Canonical Reductions................. 79
3.4.4 Other Reductions ...................... 79
3.5 Kronecker Products .............................. 79
3.5.1 Outer Products of Tableaux ......... 79
3.5.2 Other Outer Products ................. 80
3.5.3 Plethysms ................................ 80
3.6 Atomic States ...................................... 80
3.6.1 Shell Structure .......................... 80
3.6.2 Automorphisms of SO(8)............. 81
3.6.3 Hydrogen and Hydrogen-Like
Atoms ...................................... 81
3.7 The Generalized Wigner–Eckart Theorem 82
3.7.1 Operators ................................. 82
3.7.2 The Theorem............................. 82
3.7.3 Calculation of the Isoscalar
Factors..................................... 82
3.7.4 Generalizations of Angular
Momentum Theory .................... 83
3.8 Checks ................................................ 83
References .................................................. 84
3.1 Generators
3.1.1 Group Elements
An element S
a
of a Lie group G corresponding to
an infinitesimal transformation can be written in the
form
S
a
= 1 +δa
σ
X
σ
, (3.1)
where the δa
σ
are the infinitesimal parameters and
the X
σ
are the generators [3.1]. Summation over
the repeated Greek index is implied. Transformations
corresponding to finite parameters can be found by
exponentiation:
S
a
→ exp
a
1
X
1
exp
a
2
X
2
···exp
a
r
X
r
.
(3.2)
The generators necessarily form a Lie algebra, that is,
they close under commutation:
X
ρ
, X
σ
= c
τ
ρσ
X
τ
. (3.3)
Part A 3
76 Part A Mathematical Methods
In terms of the structure constants c
τ
ρσ
, the metric tensor
is defined as
g
ρσ
= c
µ
ρλ
c
λ
σµ
. (3.4)
3.1.2 Conditions on the Structure Constants
For an Abelian group, all the generators commute with
one another:
c
τ
ρσ
= 0 . (3.5)
The operators X
σ
,(σ = 1, 2,... , p < r) form the gen-
erators of a subgroup if [3.2]
c
τ
ρσ
= 0 ,(ρ,σ≤ p,τ>p). (3.6)
The subgroup is invariant if the stronger condition
c
τ
ρσ
= 0 ,(ρ≤ p,τ>p) (3.7)
is satisfied. A group is simple if it contains no in-
variant subgroup (besides the unit element). A group
is semisimple if it contains no Abelian invariant sub-
group (besides the unit element). A necessary and
sufficient condition that a group be semisimple is
that
det | g
ρτ
|= 0 . (3.8)
All simple groups are semisimple. For semisimple
groups, the inverse tensor g
µν
can be formed, thus
permitting suffixes to be raised. The quadratic operator
C = g
ρσ
X
ρ
X
σ
(3.9)
commutes with all generators of the group and is called
Casimir’s operator [3.1]. If the generators of a group G
can be broken up into two sets such that each member
of one set commutes with all the members of the other,
that is, if
c
τ
ρσ
= 0 ,(ρ≤ p,σ> p), (3.10)
then the two sets form the generators of two invariant
subgroups, H and K. The group G is the direct product
of H and K and is written as H × K.
3.1.3 Cartan–Weyl Form
By taking suitable linear combinations H
i
and E
α
of the generators X
σ
, the basic commutation rela-
tions (3.3) can be thro wn into the so-called Cartan–Weyl
form [3.1]
[H
i
, H
j
]=0 , (3.11)
[H
i
, E
α
]=α
i
E
α
, (3.12)
[E
α
, E
−α
]=α
i
H
i
, (3.13)
[E
α
, E
β
]=N
αβ
E
α+β
. (3.14)
The Roman symbols i, j,... run over an l-dimensional
space (the weight space of rank l) in which the num-
bers α
i
can be visualized as the components of the
vectors (called roots). The E
α
are shift operators, the
displacements being specified by the components of α.
The operator E
α+β
in (3.17) is to be interpreted as 0 if
α +β is not a root. The coefficient N
αβ
depends on the
choice of normalization.
3.1.4 Atomic Operators as Generators
The pairs a
†
ξ
a
η
of creation and annihilation operators
for either bosons or fermions, as defined in Sect. 6.1.1
close under commutation and form a Lie algebra. The
coupled forms W
(κk)
, defined in Sect. 6.2.2,areoften
used to play the role of the generators for electrons in an
atomic shell.
3.2 Classification of Lie Algebras
3.2.1 Introduction
The semisimple Lie algebras have been classified by
Cartan [3.3]. They consist of four main classes A
l
,
B
l
, C
l
, D
l
, and five exceptions G
2
, F
4
, E
6
, E
7
, E
8
.
Each algebra is characterized by an array of roots in
the l-dimensional weight space; they are conveniently
specified by a set of mutually orthogonal unit vec-
tors e
i
. The total number of generators (those of type
E
α
plus the l generators of type H
i
)givestheorder of
the algebra.
3.2.2 The Semisimple Lie Algebras
A
l
. The roots are conveniently represented by the vectors
e
i
−e
j
(i, j = 1, 2,... ,l+1). They are all perpendicu-
lar to Σe
k
and do not extend beyond the l-dimensional
weight space. The order of the algebra is l(l +2).The
group for which this algebra can serve as a basis is the
special unitary group SU(l +1).
B
l
. The roots are ±e
i
and ±e
i
±e
j
(i, j = 1, 2,... ,
l;i = j). The order of the algebra is l(2l+1).Acor-
Part A 3.2
Group Theory for Atomic Shells 3.3 Irreducible Representations 77
Table 3.1 Generators of the Lie groups for the atomic l shell. The subscripts i and j run over all 4l+2 states of a single
electron
Group Generators
SO(8l+5)
a
a
†
i
a
†
j
, a
†
i
a
j
, a
i
a
j
, a
†
i
, a
j
SO(8l+4)
a
a
†
i
a
†
j
, a
†
i
a
j
, a
i
a
j
U(4l+2)
b
W
(κk)
(κ = 0, 1; k = 0, 1,... ,2l)
SU(4l+2)
b
W
(κk)
(As above, with κ =k = 0 excluded)
Sp(4l+2)
c
W
(κk)
(As above, with κ +k odd)
U(2l+1)
d
W
(0k)
(k =0, 1,... ,2l)
SU(2l+1)
d
W
(0k)
(k =1, 2,... ,2l)
SO(2l+1)
d
W
(0k)
(k =1, 3, 5,... ,2l−1)
G
d
2
W
(01)
, W
(05)
(for l = 3)
SO
L
(3)
e
W
(01)
(or L)
SO
S
(3)
e
W
(10)
(or S)
U
A
(2l+1) ×U
B
(2l+1)
f
W
(0k)
0q
+W
(1k)
0q
, W
(0k)
0q
−W
(1k)
0q
(k =0, 1,... ,2l)
SO
λ
(2l+1) ×SO
µ
(2l+1) ×SO
ν
(2l+1) ×SO
ξ
(2l+1)
g
(θ
†
θ)
(k)
(k odd, θ ≡ λ, µ, ν, ξ)
U
λ
(2
l
) ×U
µ
(2
l
) ×U
ν
(2
l
) ×U
ξ
(2
l
)
h
q
†
θ
q
θ
(all components, θ ≡λ, µ, ν, ξ)
a
[3.4, 5]
b
[3.1, 6]
c
[3.6, 7]
d
[3.6]
e
[3.8]
f
[3.9]
g
[3.10]and(6.69)–(6.72)
h
[3.11]
responding group is the special orthogonal (or rotation)
group in 2l+1 dimensions, SO(2l+1).
C
l
. The roots are ±2e
i
and ±e
i
±e
j
(i, j = 1, 2,... ,
l;i = j). The order of the algebra is l (2l+1). A corres-
ponding group is the symplectic group in 2l dimensions,
Sp(2l). A rotation of the roots yields C
2
= B
2
.
D
l
. The roots are ±e
i
±e
j
(i, j = 1, 2,... ,l ;i = j).
The order of the algebra is l (2l −1). A correspond-
ing group is the special orthogonal (or rotation) group
SO(2l). A rotation of the roots yields D
3
= A
3
.Also,
D
2
= A
1
× A
1
.
E
6
, E
7
, E
8
. The roots are given by Racah [3.1]. The re-
spective orders are 78, 133, and 248.
F
4
. The roots consist of the roots of B
4
together with
the 16 vectors
1
2
( ±e
1
±e
2
±e
3
±e
4
). The order of the
algebra is 52.
G
2
. The roots consist of the roots of A
2
together with
the six vectors ±(2e
i
−e
j
−e
k
)(i = j = k = 1, 2, 3).
The order of the algebra is 14.
Examples of Lie groups used in atomic shell
theory, together with their generators, are given in Ta-
ble 3.1.
3.3 Irreducible Representations
3.3.1 Labels
If n atomic states of a collection transform among
themselves under an arbitrary action of the generators
of a group G, then the states form a representation
of G. The representation is irreducible if n
linear
combinations of the states cannot be found that also
exhibit that property, where n
< n. The commuting
generators H
i
of G can be simultaneously diagonalized
within the n states: their eigenvalues (m
1
, m
2
,... ,m
l
)
for an eigenstate ψ specify the weight of the eigen-
state. The weight above is said to be higher than
m
1
, m
2
,... ,m
l
if the first non-vanishing term in the
sequence m
1
−m
1
, m
2
−m
2
,... is positive. An irre-
ducible representation (IR) of a semisimple group is
uniquely specified (to within an equivalence) by its
highest weight [3.1], which can therefore be used as
a defining label.
3.3.2 Dimensions
The dimensions of the IRs of various groups are ex-
pressed in terms of the highest weights and set out
in Table 3.2. General algebraic expressions have been
given by Wybourne [3.12, pp. 137]. Numerical tabu-
lations have been made by Butler in the appendix to
another book by Wybourne [3.13],andalsobyMcKay
and Patera [3.14]. The latter defines the IRs by speci-
Part A 3.3