Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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426 10 Group Theory
identity element, such that
b
i
Φ
i
0,bb
i
m
j1
(Φ
i
a, b
(a
j
a0
a
j
(10.357)
The volume at b is then
b Jba (10.358)
where
Jb
Det
S
T
T
T
T
T
T
T
T
T
U
(Φ
i
a, b
(a
j
a0
V
W
W
W
W
W
W
W
W
W
X
(10.359)
is the Jacobian determinant. Let Ρ0 denote the density near the identity element, such
that
Ρbb Ρ0a Ρb
Ρ0
Jb
(10.360)
defines a suitable density function. The density at the origin is usually assigned Ρ01,
somewhat arbitrarily. This version is described as a right density because it is based upon
the right argument of c Φa, b.
Suppose that Ψa is a function of the group parameters. Recall that the rearrangement
theorem for finite groups states
g
Ψg
g
Ψgh
g
Ψhg (10.361)
where h belongs to the group and the summations are taken over all group elements g.
Generalizations to continuous groups then take the form
ΨaΡaa
Ψ
Φa, b
Ρ
R
bb
bΡ
L
bΨ
Φb, a
(10.362)
where one must recognize that left and right densities need not be equal if one or more
parameters has an infinite range. Thus, depending upon application, we may require both
Ρ
R
bΡ0
+
Det
S
T
T
T
T
T
T
T
T
T
U
(Φ
i
a, b
(a
j
a0
V
W
W
W
W
W
W
W
W
W
X
(10.363)
Ρ
L
bΡ0
+
Det
S
T
T
T
T
T
T
T
T
T
U
(Φ
i
b, a
(a
j
a0
V
W
W
W
W
W
W
W
W
W
X
(10.364)
where Ρ0 may be chosen for convenience. Fortunately, one can show that these densities
are equal for compact groups and we usually will not have to confront this issue.
Perhaps the most important application of group integration is to the orthogonality
properties of irreducible representations whose matrix elements are analytic functions of
10.6 Orthogonality Relations for Lie Groups 427
the continuous group parameters. Recall that the orthogonality theorem for two irreps of a
finite group with N
G
elements takes the form
gG
D
i
Μ,Ν
g
D
j
Α,Β
g
N
G
n
i
∆
Α,Μ
∆
Β,Ν
∆
i, j
(10.365)
where n
i
is the dimensionality of irrep i. This result can now be translated to the continuum
according to
D
i
Μ,Ν
a
D
j
Α,Β
aΡaa
∆
Α,Μ
∆
Β,Ν
∆
i, j
n
i
Ρaa (10.366)
where the integral over density is the analog of N
G
. Naturally we assume that the integrals
exist but will not investigate the necessary conditions; suffice it to say that problems are
rarely encountered here. Similarly, orthogonality of characters
k
N
k
Χ
i
k
Χ
j
k
N
G
∆
i, j
(10.367)
takes the continuum form
Χ
i
a
Χ
j
aa ∆
i, j
Ρaa (10.368)
Example
For example, consider
x
'
a
0
1 a
1
x, a
1
1 (10.369)
such that
Φ
0
a, ba
0
b
0
1 a
1
(Φ
0
(a
0
1,
(Φ
0
(a
1
b
0
(10.370)
Φ
1
a, ba
1
b
1
1 a
1
(Φ
1
(a
0
0,
(Φ
1
(a
1
1 b
1
(10.371)
results in
Jb1 b
1
(10.372)
such that
Ρb
1
1 b
1
(10.373)
428 10 Group Theory
10.7 Quantum Mechanical Representations of the Rotation Group
10.7.1 Generators and Commutation Relations
We have seen in previous sections that the wave functions for spin-
1
2
transform among
themselves according to SU(2) while those for orbital angular momentum are governed by
SO(3). Both of those groups have dimension 3, rank 1, and satisfy the same commutation
relations. Therefore, we generalize the definition of angular momentum to include both
spin and orbital angular momentum by postulating commutation relations of the form
J
i
,J
j
i, j,k
J
k
(10.374)
for the generators of rotations of the wave function and define the square of the total
angular momentum as the Casimir operator
J
2
J
2
x
J
2
y
J
2
z
J
2
,J
i
0 (10.375)
that commutes with all of the generators. Therefore, we can define basis vectors Ψ
j,m
that
are simultaneous eigenfunctions of J
2
and one of its components, arbitrarily chosen as J
z
,
such that
J
z
Ψ
j,m
mΨ
j,m
(10.376)
It is also useful to define raising and lowering operators, also known as ladder operators ,
according to
J
J
x
J
y
(10.377)
such that
J
z
,J
J
(10.378)
J
,J
2
0 (10.379)
J
,J
2J
z
(10.380)
Thus,
J
z
J
J
J
z
Ψ
j,m
J
Ψ
j,m
J
z
J
Ψ
j,m
m 1J
Ψ
j,m
(10.381)
J
z
J
J
J
z
Ψ
j,m
J
Ψ
j,m
J
z
J
Ψ
j,m
m 1J
Ψ
j,m
(10.382)
demonstrates that J
Ψ
j,m
is an eigenfunction of J
z
with eigenvalue m 1–theeffect of J
is to raise m by one unit whereas J
lowers m by the same amount. Suppose that Μ is the
maximum possible m in the irreducible representation labeled j. Expressing J
2
in the form
J
2
J
J
J
z
J
z
1
(10.383)
we obtain
ΜMaxmJ
Ψ
j,Μ
0 J
2
Ψ
j,Μ
ΜΜ1Ψ
j,Μ
(10.384)
10.7 Quantum Mechanical Representations of the Rotation Group 429
and can label the representation by its maximum m such that
J
2
Ψ
j,m
j j 1Ψ
j,m
(10.385)
for any allowed m. Application of
J
n
to Ψ
j,j
yields an eigenfunction Ψ
j,jn
and must
terminate at a finite value of n because the representation j is finite. Expressing J
2
in the
form
J
2
J
J
J
z
J
z
1
(10.386)
and using J
Ψ
j,jn
0, we obtain
J
2
Ψ
j,jn
j n j n 1Ψ
j,jn
j j 1Ψ
j,jn
(10.387)
where the final step uses the fact that the eigenvalue of J
2
is independent of m.Thus,we
find j n/2 where n is an integer, and conclude that j is either integral or half-integral
according to whether n is even or odd. The operators J
, J
z
transform the eigenfunctions of
J
2
with common j among themselves and must span the vector space because the represen-
tation we seek is irreducible. Therefore, the irreducible representation labeled j contains
2 j 1 basis vectors that satisfy
J
2
Ψ
j,m
j j 1Ψ
j,m
(10.388)
J
z
Ψ
j,m
mΨ
j,m
(10.389)
for m j, j 1,..., j 1,j.
All rotations through the same angle, regardless of axis, are similar and belong to
the same class. Therefore, there are an infinite number of classes and, correspondingly, an
infinite number of irreducible representations. To demonstrate this class structure, consider
a rotation S R
x
ΘR
z
Φ that creates a new coordinate system with its polar axis in the
direction Θ, Φ wrt to the original coordinate system. If we follow a rotation about this
new axis by S
1
to restore the original coordinate system, the net result of SR
z
ΑS
1
is to
perform a rotation through angle Α about an axis specified by Θ, Φ. This general rotation
is similar to R
z
Α by construction. Thus, all rotations through the same angle around any
axis are similar to each other and form a class. The irreducible representations for rotations
about the z-axis take the form
D
j
R
z
Θ
ExpmΘ (10.390)
with character
Χ
j
Θ Tr ExpmΘ
j
m j
ExpmΘ
Sin2 j 1Θ/2
SinΘ/ 2
(10.391)
where Θ labels the class. More generally, an arbitrary rotation is represented by
D
j
Ω ExpΩ , J (10.392)
where ΩΘ
x
, Θ
y
, Θ
z
.TheD
j
matrices are unitary, which requires the J
i
operators to be
hermitian.
430 10 Group Theory
Finally, we evaluate the matrix elements of J
.Let
J
Ψ
j,m
a
j,m
Ψ
j,m1
(10.393)
J
Ψ
j,m
b
j,m
Ψ
j,m1
(10.394)
and observe that these operators are hermitian conjugates of each other
J
J
,J
J
(10.395)
Furthermore, the cross-products take the form
J
J
J
2
J
z
J
z
1
(10.396)
J
J
J
2
J
z
J
z
1
(10.397)
Thus, the norms
a
j,m
2
'
J
Ψ
j,m
J
Ψ
j,m
(
Ψ
j,m
J
J
Ψ
j,m
j j 1mm 1 (10.398)
b
j,m
2
'
J
Ψ
j,m
J
Ψ
j,m
(
Ψ
j,m
J
J
Ψ
j,m
j j 1mm 1 (10.399)
determine the matrix elements up to phase. Choosing the so-called Condon–Shortley con-
ventions
J
Ψ
j,m
j j 1mm 1Ψ
j,m1
(10.400)
J
Ψ
j,m
j j 1mm 1Ψ
j,m1
(10.401)
then determines the relative phases among the Ψ
j,m
basis vectors. Notice that J
Ψ
j,j
J
Ψ
j,j
0, as required by the preceding analysis.
10.7.2 Euler Parametrization
The orientation of a three-dimensional object can be specified by three Euler angles
describing the following sequence of operations:
• rotation about the ˆz-axis through angle Α, producing new axes ˆx
'
, ˆy
'
, ˆz
'
ˆz
• rotation about the ˆy
'
-axis through angle Β, producing new axes ˆx
''
, ˆy
''
ˆy
'
, ˆz
''
• rotation about the ˆz
''
-axis through angle Γ.
Alternatively, the same orientation can be achieved using the following sequence opera-
tions with respect to space-fixed axes:
• rotation about the ˆz-axis through angle Γ
• rotation about the ˆy-axis through angle Β
• rotation about the ˆz-axis through angle Α.
10.7 Quantum Mechanical Representations of the Rotation Group 431
Thus, a general rotation matrix takes the form
RΑ, Β, Γ R
z
ΑR
y
ΒR
z
Γ (10.402)
with inverses
RΑ, Β, Γ
1
R
z
ΓR
y
ΒR
z
Α RΓ, Β, Α (10.403)
The ranges for these angles are 0 Α2Π,0ΒΠ, and 0 Γ2Π. However, a spec-
ified rotation does not uniquely determine the Euler angles – for example, the parameters
Α, 0, Γ and Α Κ, 0, ΓΚproduce the same rotation for any Κ.
Suppose that Ψ
j,m
is an eigenfunction with total angular momentum j and magnetic
quantum number m.Theeffect of a rotation is then
Α, Β, ΓΨ
j,m
j
m
'
j
Ψ
j,m
'
D
j
m
'
m
Α, Β, Γ (10.404)
where D
j
is the rotation matrix for irreducible representation j. Please note once again
the ordering of the indices. Having chosen basis functions to be eigenfunctions of J
z
,we
can write
D
j
Α, Β, Γ Exp
ΑJ
z
d
j
Β Exp
ΓJ
z
(10.405)
where the reduced rotation matrix d
j
depends only upon Β while the rotations about the
z-axis are diagonal. Thus,
Α, Β, ΓΨ
j,m
j
m
'
j
Ψ
j,m
'
d
j
m
'
m
Β Exp
mΓm
'
Α
(10.406)
10.7.3 Homomorphism Between SU(2) and SO(3)
The groups SO(3) and SU(2) both require three real parameters and their generators satisfy
the same commutation relations, suggesting that these groups are homomorphic. There are
two common parametrizations of SU(2). First, consider the Pauli representation
U
p
0
p
3
p
2
p
1
p
2
p
1
p
0
p
3
p
0
Σ
0
p ,
Σ (10.407)
where the parameters p
i
are real,
Σ
0
10
01
, Σ
1
01
10
, Σ
2
0
0
, Σ
3
10
0 1
(10.408)
are the basis matrices, and where the constraint
p
2
0
1 p
2
1
p
2
2
p
2
3
(10.409)
432 10 Group Theory
ensures that DetU1. Note that there are only three independent parameters and that
we require
p1. Alternatively, the Cayley–Klein parametrization
UΞ, Ψ, Ζ
Ζ
CosΨ
Ξ
SinΨ
Ξ
SinΨ
Ζ
CosΨ
(10.410)
has the advantage that the parameters are unconstrained and U is manifestly unitary.
Consider the matrix
Ρ
r ,
Σ
zxy
x y z
(10.411)
from which one can read the components of
r in a particular coordinate system using
x
Ρ
21
Ρ
12
*
2 (10.412)
y
Ρ
21
Ρ
12
*
2 (10.413)
z Ρ
11
(10.414)
Next, suppose that we evaluate
Ρ
'
UΡU
1
(10.415)
with a similarity transform and evaluate the new coordinates
r
'
A
r (10.416)
by interpreting the elements of Ρ
'
in the same manner. The resulting linear transformation
is represented by the 3 3matrixA.IfA is an orthogonal matrix with determinant 1, it
can be interpreted as a coordinate rotation that establishes a homomorphism between the
SU(2) matrix U and the SO(3) matrix A. After some straightforward but tedious algebra
(see exercises), one finds
A
3
4
4
4
4
4
4
4
4
4
4
4
5
1 2
p
2
2
p
2
3
2
p
0
p
3
p
1
p
2
2
p
1
p
3
p
0
p
2
2
p
1
p
2
p
0
p
3
1 2
p
2
1
p
2
3
2
p
2
p
3
p
0
p
1
2
p
0
p
2
p
1
p
3
2
p
2
p
3
p
0
p
1
1 2
p
2
1
p
2
2
6
7
7
7
7
7
7
7
7
7
7
7
8
(10.417)
and can verify that AA
T
and DetA1usingp
2
0
1 p
2
1
p
2
2
p
2
3
. Therefore,
A SO(3) is a valid rotation matrix.
Next we seek the relationship between the Euler angles and the Cayley–Klein parame-
ters. Although one can evaluate Ρ
'
UΡU
1
using the Cayley–Klein parametrization in
general, it is easier to choose the following special cases:
U0, 0, Ζ
Ζ
0
0
Ζ
UΡU
1
3
4
4
4
4
4
4
4
5
Cos2Ζ Sin2Ζ 0
Sin2Ζ Cos2Ζ 0
001
6
7
7
7
7
7
7
7
8
R
z
2Ζ (10.418)
U0, Ψ, 0
CosΨ SinΨ
SinΨ CosΨ
UΡU
1
3
4
4
4
4
4
4
4
5
Cos2Ψ 0Sin2Ψ
010
Sin2Ψ 0Cos2Ψ
6
7
7
7
7
7
7
7
8
R
y
2Ψ
(10.419)
10.7 Quantum Mechanical Representations of the Rotation Group 433
Thus, we have expressed the two matrices needed for the Euler parametrization of SO(3)
in terms of the Cayley–Klein parametrization of SU(2). A general rotation then takes the
form
RΑ, Β, Γ R
z
ΑR
y
ΒR
z
Γ U U
0, 0,
Α
2
U
0,
Β
2
, 0
U
0, 0,
Γ
2
(10.420)
where the reflects the sign ambiguity in U that arises because R was derived from
UΡU
1
, which is second-order in U. This homomorphism is double-valued: two elements
of SU(2) map onto the same element of SO(3).
Transformation matrices for state vectors with j 1/ 2 are, as usual, based upon the
inverse of the rotation for coordinates
RΑ, Β, Γ
1
R
z
ΓR
y
ΒR
z
Α
U
1
U
0, 0,
Γ
2
U
0,
Β
2
, 0
U
0, 0,
Α
2
(10.421)
such that
D
1/ 2
Α, Β, Γ
ΑΓ/ 2
Cos
Β
2
ΑΓ/ 2
Sin
Β
2
ΓΑ/ 2
Sin
Β
2
ΑΓ/ 2
Cos
Β
2
(10.422)
is a two-dimensional irreducible representation of SO(3) that describes states with angular
momentum j 1/ 2. It is also useful to factor the rotation matrix
D
1/ 2
Α, Β, Γ Exp
Γ Σ
3
*
2
d
1/ 2
Β Exp
Α Σ
3
*
2
(10.423)
where the reduced rotation matrix is simply
d
1/ 2
Β
Cos
Β
2
Sin
Β
2
Sin
Β
2
Cos
Β
2
(10.424)
Notice that a rotation of the coordinate system by 2Π around any axis simply inverts the
sign of the wave function; a rotation through 4Π is needed to recover the original wave
function. The half-angles appear here because the homomorphism between SU(2) and
SO(3) is double-valued. Despite this minor complication, this representation for D
1/ 2
provides a useful starting point for construction of irreducible representations of higher
dimension.
10.7.4 Irreducible Representations of SU(2)
Consider two variables Χ
and Χ
and construct linear differential operators
J
x
1
2
Χ
(
(Χ
Χ
(
(Χ
(10.425)
J
y
2
Χ
(
(Χ
Χ
(
(Χ
(10.426)
J
z
1
2
Χ
(
(Χ
Χ
(
(Χ
(10.427)
434 10 Group Theory
that act on functions of the form ΨΧ
, Χ
. With careful examination of commutators like
J
x
J
y
J
y
J
x
Ψ
4
Χ
(
(Χ
Χ
(
(Χ
Χ
(Ψ
(Χ
Χ
(Ψ
(Χ
Χ
(
(Χ
Χ
(
(Χ
Χ
(Ψ
(Χ
Χ
(Ψ
(Χ
4
Χ
(Ψ
(Χ
Χ
Χ
(
2
Ψ
(Χ
(Χ
Χ
2
(
2
Ψ
(Χ
2
Χ
2
(
2
Ψ
(Χ
2
Χ
(Ψ
(Χ
Χ
Χ
(
2
Ψ
(Χ
(Χ
4
Χ
(Ψ
(Χ
Χ
Χ
(
2
Ψ
(Χ
(Χ
Χ
2
(
2
Ψ
(Χ
2
Χ
2
(
2
Ψ
(Χ
2
Χ
(Ψ
(Χ
Χ
Χ
(
2
Ψ
(Χ
(Χ
2
Χ
(Ψ
(Χ
Χ
(Ψ
(Χ
J
z
Ψ
(10.428)
one soon verifies that these operators satisfy the commutation relations
J
i
,J
j
ijk
J
k
(10.429)
required for angular momentum operators. Next, suppose that Ψ assumes the specific form
Ψ
j,m
2 j!
j m! j m!
Χ
jm
Χ
jm
(10.430)
of homogeneous polynomials of order 2 j where m j, j 1,..., j 1,jand where the
normalization factor is chosen to ensure that
j
m j
Ψ
j,m
2
j
m j
2 j
j m
Χ
jm
Χ
jm
2
Χ
2
Χ
2
j
1 (10.431)
when the basis vectors are normalized according to
Χ
2
Χ
2
1. Then using
Χ
(
(Χ
Ψ
j,m
j mΨ
j,m
, Χ
(
(Χ
Ψ
j,m
j mΨ
j,m
J
z
Ψ
j,m
mΨ
j,m
(10.432)
we observe that Ψ
j,m
is an eigenfunction of J
z
with eigenvalue m. Similarly, using
J
2
J
2
x
J
2
y
J
2
z
1
4
Χ
2
(
2
(Χ
2
Χ
2
(
2
(Χ
2
2Χ
Χ
(
2
(Χ
(Χ
3Χ
(
(Χ
3Χ
(
(Χ
(10.433)
10.7 Quantum Mechanical Representations of the Rotation Group 435
such that
J
2
Ψ
j,m
1
4
j m j m 1j m j m 12j m j m3 j m
3 j m
Ψ
j,m
j j 1Ψ
j,m
(10.434)
we find that Ψ
j,m
is an eigenfunction of J
2
with eigenvalue j j 1. Therefore, Ψ
j,m
con-
stitutes a suitable set of basis functions for representations of the rotation group.
We now require the variables Χ
, Χ
to transform under SU(2) according to D
1/ 2
Α, Β, Γ,
such that
Χ
'
m
1/ 2
m
'
1/ 2
Χ
m
'
D
1/ 2
m
'
,m
Α, Β, Γ (10.435)
where Χ
1/ 2
is equivalent to Χ
.Itissufficient to work out the case ΑΓ0, for which
Χ
'
jm
Χ
'
jm
Cos
Β
2
Χ
Sin
Β
2
Χ
jm
Sin
Β
2
Χ
Cos
Β
2
Χ
jm
jm
Μ0
jm
Ν0
Μ
j m
Μ
j m
Ν
Cos
Β
2
jmΜΝ
Sin
Β
2
jmΜΝ
Χ
2 jΜΝ
Χ
ΜΝ
(10.436)
where the second line arises from the binomial theorem. The terms under the summation
can be expressed in terms of Ψ
j,m
'
using the substitution ΜΝ! j m
'
such that
Χ
'
jm
Χ
'
jm
j
m
'
j
Ν
jm
'
Ν
j m
j m
'
Ν
j m
Ν
Cos
Β
2
mm
'
2Ν
Sin
Β
2
2 jmm
'
2Ν
Χ
jm
'
Χ
jm
'
(10.437)
where the summation over Ν is limited by the binomial coefficients, which vanish when
the lower argument is negative. Upon insertion of the normalization factors
Ψ
'
j,m
j
m
'
j
Ν
jm
'
Ν
j m
'
!
j m
'
!
j m! j m!
1/ 2
j m
j m
'
Ν
j m
Ν
Cos
Β
2
mm
'
2Ν
Sin
Β
2
2 jmm
'
2Ν
Ψ
j,m
'
(10.438)