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  Y lm T d  Bloch functions Operators of system can be classified in terms of IRs x, p transforms under rotations as vectors T , F  as 2 nd 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). { e j  | j = 1, …, n  } is an orthonormal basis of V .  g  G where D  is the matrix IR wrt { e j  } Definition 4.1: Irreducible Basis Vectors ( IBV ) { e j  } is an irreducible set transforming according to the  –rep of G. e j  is said to belong to the j th row of the  –rep.

Generalization of the Orthogonality Theorem: Let IR  and be equivalent, i.e.,  S  D = S D  S  1. Thus, the Orthogonality Theorem can be generalized to where

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

Let → Comparing with gives i.e.,  

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, is a set of IBVs that transform according to  , if not null, Proof: QED (  ) exempts  from sum rule

Theorem 4.3: Letbe a set of IBVs & Then Proof: QED If D  is orthogonal, then

Corollary 1: Proof: is a set of (un-normalized) IBVs Alternative proof:

Corollary 2: Proof: { e j  } is complete:  QED The matrix elements of U(g) wrt the IBVs are Hence ( Block diagonal  g )

Corollary 3: Proof: Cor. 2: ( Cor. 1 ) QED C.f. Proof of Theorem 4.2

Corollary 3: Alternative proof: QED

Definition 4.2: Projection Operators = Projection operator onto basis vector e  j = Projection operator onto irred invariant subspace V   P  j & P  are indeed projections Or:

Theorem 4.4: Completeness P  j & P  are complete, i.e., Proof:Letbe the basis of any irreducible invariant subspace V of V Thm 4.3: , k QED Alternatively: ( Provided { e  j } is complete, i.e., V is decomposible. )

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 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, I S }, where I S x = –x. G  C 2 eISIS  1 11  2 1–1 For 1–D reps:

Problem 3.9 Due date:Monday, Apr 10. Mid-Term Take-Home Exam

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

Example 3: NH 3

1. Transform a basis to IBVs. E.g., From localized basis to IBVs( normal modes ) Time dependence of normal modes are harmonic Applications 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 where = reduced matrix element Proof:Thm 4.1: QED sum over  See Tinkham for a more detailed proof.

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

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