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

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

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

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)

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

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

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

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

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

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:

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

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

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

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

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) :

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