1. 4. General Properties of Irreducible Vectors and Operators
4.1 Irreducible Basis Vectors
4.2 The Reduction of Vectors — Projection Operators for Irreducible Components
4.3 Irreducible Operators and the Wigner-Eckart Theorem
Comments:
States of system can be classified in terms of IRs
Spherical symmetry  Ylm
Td  Bloch functions
Operators of system can be classified in terms of IRs
x, p transforms under rotations as vectors
T , F as 2nd rank tensors under Lorentz transformations

2. 4.1. Irreducible Basis Vectors
Notations:
U(G) is an unitary rep of G on an inner product space V.
V is an invariant subspace of V wrt U(G).
{ ej | j = 1, …, n } is an orthonormal basis of V. 
 gG
where D is the matrix IR wrt { ej }
Definition 4.1: Irreducible Basis Vectors ( IBV )
{ ej } is an irreducible set transforming according to the –rep of G

3. Theorem 4.1:
Let { uj | j = 1, …, n } & { vk | k = 1, …, n } be 2 IBVs wrt G on V.
If  &  are inequivalent, then { uj } & { vk } are mutually orthogonal.
Proof:
 gG
QED

4. If  &  are equivalent, then
If span{ ej }  span{ ek } = { 0 }, then
Otherwise, span{ ej } = span{ ek } &
where S is unitary
Example: H-atom, G = R(3)

5. 4.2. The Reduction of Vectors – Projection Operators for Irreducible Components
Theorem 4.2:
Let
Then for any | x   V,
, if not null,
is a set of IBVs that transform according to 
Proof:
QED
(  ) exempts  from sum rule

6. Theorem 4.3:
Let
be a set of IBVs &
Then
Proof:
Corollary 1:
Proof:
is a set of IBVs

7. Corollary 2:
Proof: This is just the inverse of the defining eq of P in Theorem 4.2
Corollary 3:
Proof:
Cor. 2:
( Cor. 1 )
QED

8. Definition 4.2: Projection Operators
= Projection operator onto basis vector
= Projection operator onto irreducible invariant subspace V
 P j & P are indeed projections
Theorem 4.4: Completeness
P j & P are complete, i.e.,
Proof:
Let
be the basis of any irreducible invariant subspace V of V
Thm 4.3:
 , k
QED

9. Comments:
Let U(G) be a rep of G on V.
If U(G) is decomposable, then
The corresponding complete set of IBVs is
Then
is not exactly a projection, but it's useful in constructing IBVs

10. Example 1:
Let V be the space of square integrable functions f(x) of 1 variable.
Let G = { e, IS } , where IS x = –x.
G  C2 e IS
1 1
2 –1
For 1–D reps:

11. Example 2: Td = { T(n) | nZ }
G = Td. V = Space of state vectors for a particle on a 1–D lattice.
IR :
b = lattice constant
Let | y  be any localized states in the unit cell
 | k, y  is an eigenstate of T(m) with eigenvalue e– i k m b (c.f. Chap 1)
( State periodic )
( Prob 4.1 )
 All distinct IBVs can be generated from | y  in the unit cell

12. Applications 1. Transform a basis to IBVs. E.g., From localized basis
( normal modes )
Time dependence of normal modes are harmonic
2. Reduce direct product reps to IRs & evaluate C-GCs
Prob 4.2

13. 4.3. Irreducible Operators and the Wigner-Eckart Theorem
Definition 4.3: Irreducible Operators ( tensors )
Operators { Oj | j = 1, …, n } are irreducible corresponding to the IR  if
Comments:
Let { Oj } & { ej } be irreducible. Then
i.e., Oj e k transforms according to D 
implicit sum

14. Theorem 4.5 Wigner-Eckart
Let { Oj } & { ej } be irreducible. Then
sum over 
where
= reduced matrix element
Proof:
Thm 4.1:
QED

15. Example: EM Transitions in Atoms, G = R(3)
Atom: | j m : m = –j, –j+1, …, j–1, j
Photon ( s,  ): s = 1,  = –1, 0, +1
Transition rate W  | f |2
Os = dipole operator
Wigner-Eckart ( = 1) :

16. Allowed transitions with branching ratios ( Inversion not considered )
Transition w/o symmetry considerations j = j' = 1