Introduction to Group Theory With Applications to Quantum Mechanics and Solid State Physics Roland Winkler
[email protected] August 2011 (Lecture notes version: November 3, 2015) Please, let me know if you find misprints, errors or inaccuracies in these notes. Thank you.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
General Literature I
J. F. Cornwell, Group Theory in Physics (Academic, 1987) general introduction; discrete and continuous groups
I
W. Ludwig and C. Falter, Symmetries in Physics (Springer, Berlin, 1988). general introduction; discrete and continuous groups
I
W.-K. Tung, Group Theory in Physics (World Scientific, 1985). general introduction; main focus on continuous groups
I
L. M. Falicov, Group Theory and Its Physical Applications (University of Chicago Press, Chicago, 1966). small paperback; compact introduction
I
E. P. Wigner, Group Theory (Academic, 1959). classical textbook by the master
I
Landau and Lifshitz, Quantum Mechanics, Ch. XII (Pergamon, 1977) brief introduction into the main aspects of group theory in physics
I
R. McWeeny, Symmetry (Dover, 2002) elementary, self-contained introduction
I
and many others Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Specialized Literature I
G. L. Bir und G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, New York, 1974) thorough discussion of group theory and its applications in solid state physics by two pioneers
I
C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids (Clarendon, 1972) comprehensive discussion of group theory in solid state physics
I
G. F. Koster et al., Properties of the Thirty-Two Point Groups (MIT Press, 1963) small, but very helpful reference book tabulating the properties of the 32 crystallographic point groups (character tables, Clebsch-Gordan coefficients, compatibility relations, etc.)
I
A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, 1960) comprehensive discussion of the (group) theory of angular momentum in quantum mechanics
I
and many others Roland Winkler, NIU, Argonne, and NCTU 2011−2015
These notes are dedicated to Prof. Dr. h.c. Ulrich R¨ ossler from whom I learned group theory R.W.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Introduction and Overview Definition: Group A set G = {a, b, c, . . .} is called a group, if there exists a group multiplication connecting the elements in G in the following way (1)
a, b ∈ G :
(2)
a, b, c ∈ G :
(3)
∃e ∈ G :
(4)
∀a ∈ G
c = ab ∈ G
(closure)
(ab)c = a(bc) ae = a
∃b ∈ G :
(associativity)
∀a ∈ G a b = e,
(identity / neutral element)
i.e., b ≡ a
−1
(inverse element)
Corollaries (a)
e −1 = e
(b) a−1 a = a a−1 = e (c)
e a = ae = a
(d) ∀a, b ∈ G :
∀a ∈ G
(left inverse = right inverse)
∀a ∈ G c = ab ⇔ c
(left neutral = right neutral) −1
=b
−1 −1
a
Commutative (Abelian) Group (5)
∀a, b ∈ G :
ab = ba
(commutatitivity)
Order of a Group = number of group elements Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Examples I
integer numbers
Z with addition
(Abelian group, infinite order) I
rational numbers
Q\{0} with multiplication
(Abelian group, infinite order) I
complex numbers {exp(2πi m/n) : m = 1, . . . , n} with multiplication (Abelian group, finite order, example of cyclic group)
I
invertible (= nonsingular) n × n matrices with matrix multiplication (nonabelian group, infinite order, later important for representation theory!)
I
permutations of n objects: Pn (nonabelian group, n! group elements)
I
symmetry operations (rotations, reflections, etc.) of equilateral triangle ≡ P3 ≡ permutations of numbered corners of triangle – more later!
I
(continuous) translations in ≡ vector addition in
I
R
n
Rn : (continuous) translation group
symmetry operations of a sphere
only rotations: SO(3) = special orthogonal group in R3 = real orthogonal 3 × 3 matrices Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics Group theory is the natural language to describe symmetries of a physical system I
symmetries correspond to conserved quantities
I
symmetries allow us to classify quantum mechanical states • representation theory • degeneracies / level splittings
I
evaluation of matrix elements ⇒ Wigner-Eckart theorem e.g., selection rules: dipole matrix elements for optical transitions
I
ˆ must be invariant under the symmetries Hamiltonian H of a quantum system ˆ via symmetry arguments ⇒ construct H
I
...
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Classical Mechanics I
I
I
˙ Lagrange function L(q, q), ∂L d ∂L = Lagrange equations dt ∂ q˙ i ∂qi If for one j :
∂L ∂L = 0 ⇒ pj ≡ ∂qj ∂ q˙ j
i = 1, . . . , N is a conserved quantity
Examples I
qj linear coordinate • translational invariance • linear momentum pj = const. • translation group
I
qj angular coordinate • rotational invariance • angular momentum pj = const. • rotation group
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Quantum Mechanics (1) Evaluation of matrix elements I
Consider particle in potential V (x) = V (−x) even
I
two possiblities for eigenfunctions ψ(x) ψe (x) even:
I I
ψe (x) = ψe (−x)
ψo (x) odd: ψo (x) = −ψo (−x) R ∗ ψi (x) ψj (x) dx = δij overlapp i, j ∈ {e, o} R ∗ expectation value hi|x|ii = ψi (x) x ψi (x) dx = 0
well-known explanation I
product of two even / two odd functions is even
I
product of one even and one odd function is odd
I
integral over an odd function vanishes
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Quantum Mechanics (1) Evaluation of matrix elements (cont’d) Group theory provides systematic generalization of these statements I
representation theory ≡ classification of how functions and operators transform under symmetry operations
I
Wigner-Eckart theorem ≡ statements on matrix elements if we know how the functions and operators transform under the symmetries of a system
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Quantum Mechanics (2) Degeneracies of Energy Eigenvalues I
ˆ = E ψ or i~∂t ψ = Hψ ˆ Schr¨ odinger equation Hψ
I
ˆ with i~∂t O ˆ = [O, ˆ H] ˆ =0 Let O
ˆ is conserved quantity ⇒ O
ˆ = E ψ and Oψ ˆ = λˆ ψ ⇒ eigenvalue equations Hψ O can be solved simultaneously ˆ is good quantum number for ψ ⇒ eigenvalue λOˆ of O
Example: H atom I
2 ˆ = ~ H 2m
∂2 2 ∂ + 2 ∂r r ∂r
+
ˆ2 L e2 − 2 2mr r
⇒ group SO(3)
ˆ2 , H] ˆ = [L ˆz , H] ˆ = [L ˆ2 , L ˆz ] = 0 ⇒ [L ⇒ eigenstates ψnlm (r):
ˆ2 , index l ↔ L
ˆz m↔L
I
really another example for representation theory
I
degeneracy for 0 ≤ l ≤ n − 1: dynamical symmetry (unique for H atom) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Quantum Mechanics (3) Solid State Physics in particular: crystalline solids, periodic assembly of atoms ⇒ discrete translation invariance (i) Electrons in periodic potential V (r) I
V (r + R) = V (r) ∀ R ∈ {lattice vectors}
ˆR : ⇒ translation operator T
ˆR f (r) = f (r + R) T ˆR , H] ˆ =0 [T
⇒ Bloch theorem ψk (r) = eik·r uk (r) with uk (r + R) = uk (r) ⇒ wave vector k is quantum number for the discrete translation invariance, k ∈ first Brillouin zone
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Quantum Mechanics (3) Solid State Physics rotation
(ii) Phonons
by 90 o
I
Consider square lattice
I
frequencies of modes are equal
I
degeneracies for particular propagation directions
(iii) Theory of Invariants I
How can we construct models for the dynamics of electrons or phonons that are compatible with given crystal symmetries?
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Physics
Quantum Mechanics (4) Nuclear and Particle Physics Physics at small length scales: strong interaction Proton mp = 938.28 MeV rest mass of nucleons almost equal ∼ degeneracy Neutron mn = 939.57 MeV I
Symmetry: isospin ˆI with
I
SU(2):
proton
| 12
1 2 i,
ˆ strong ] = 0 [ˆI , H neutron
| 12 − 12 i
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Mathematical Excursion: Groups Basic Concepts Group Axioms: see above Definition: Subgroup Let G be a group. A subset U ⊆ G that is itself a group with the same multiplication as G is called a subgroup of G. Group Multiplication Table: compilation of all products of group elements ⇒ complete information on mathematical structure of a (finite) group Example: permutation group P3 1 2 3 1 2 3 1 e= a= b= 1 2 3 2 3 1 3 1 2 3 1 2 3 1 c= d= f = 1 3 2 3 2 1 2 I
2 3 1 2
2 3 1 3
P3 e a b c d f
e e a b c d f
a a b e d f c
b b e a f c d
c c f d e b a
d d c f a e b
f f d c b a e
{e}, {e, a, b}, {e, c}, {e, d}, {e, f }, G are subgroups of G Roland Winkler, NIU, Argonne, and NCTU 2011−2015
I
P3 e a b Symmetry w.r.t. main diagonal c ⇒ group is Abelian d order n of g ∈ G: smallest n > 0 with g n = e f
I
{g , g 2 , . . . , g n = e} with g ∈ G is Abelian subgroup (a cyclic group)
I
in every row / column every element appears exactly once because:
Conclusions from Group Multiplication Table I
e e a b c d f
a a b e d f c
b b e a f c d
c c f d e b a
d d c f a e b
f f d c b a e
Rearrangement Lemma: for any fixed g 0 ∈ G, we have G = {g 0 g : g ∈ G} = {gg 0 : g ∈ G} i.e., the latter sets consist of the elements in G rearranged in order. proof: g1 6= g2 ⇔ g 0 g1 6= g 0 g2
∀g1 , g2 , g 0 ∈ G
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Goal: Classify elements in a group
(1) Conjugate Elements and Classes I
Let a ∈ G. Then b ∈ G is called conjugate to a if ∃ x ∈ G with b = x ax −1 . Conjugation b ∼ a is equivalence relation: • a∼a reflexive • b∼a ⇔ a∼b symmetric • a ∼ co ⇒ a∼b transitive b∼c
I
a = xcx −1 ⇒ c = x −1 ax b = ycy −1 = (xy −1 )−1 a(xy −1 )
For fixed a, the set of all conjugate elements C = {x ax −1 : x ∈ G} is called a class. • identity e is its own class
xex
−1
=e
∀x ∈ G
• Abelian groups: each element is its own class
x ax −1 = ax x −1 = a
∀a, x ∈ G
• Each b ∈ G belongs to one and only one class
⇒ decompose G into classes • in broad terms: “similar” elements form a class
Example: x e a e e a a e a b e a c e b d e b f e b
P3 b b b b a a a
c c d f c f d
d d f c f d c
f f c d d c f
⇒ classes {e}, {a, b}, {c, d, f }
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Goal: Classify elements in a group
(2) Subgroups and Cosets I
I
Let U ⊂ G be a subgroup of G and x ∈ G. The set x U ≡ {x u : u ∈ U} (the set Ux) is called the left coset (right coset) of U. In general, cosets are not groups. If x ∈ / U , the coset x U lacks the identity element: suppose ∃u ∈ U with xu = e ∈ x U ⇒ x −1 = u ∈ U ⇒ x = u −1 ∈ U
I I
If x 0 ∈ x U, then x 0 U = x U any x 0 ∈ x U can be used to define coset x U If U contains s elements, then each coset also contains s elements (due to rearrangement lemma).
I
Two left (right) cosets for a subgroup U are either equal or disjoint (due to rearrangement lemma).
I
I I
I
Thus: decompose G into cosets G = U ∪ x U ∪ y U ∪ . . . x, y , . . . ∈ /U h Thus Theorem 1: Let h order of G ⇒ ∈N Let s order of U ⊂ G s Corollary: The order of a finite group is an integer multiple of the orders of its subgroups. Corollary: If h prime number ⇒ {e}, G are the only subgroups ⇒ G is isomorphic to cyclic group Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Goal: Classify elements in a group
(3) Invariant Subgroups and Factor Groups connection: classes and cosets I
A subgroup U ⊂ G containing only complete classes of G is called invariant subgroup (aka normal subgroup).
I
Let U be an invariant subgroup of G and x ∈ G ⇔ x Ux −1 = U
⇔ x U = Ux I
(left coset = right coset)
Multiplication of cosets of an invariant subgroup U ⊂ G: x, y ∈ G :
(x U) (y U) = xy U = z U
where
z = xy
well-defined: (x U) (y U) = x (U y ) U = xy U U = z U U = z U I
An invariant subgroup U ⊂ G and the distinct cosets x U form a group, called factor group F = G/U • group multiplication: see above • U is identity element of factor group • x −1 U is inverse for x U
I
Every factor group F = G/U is homomorphic to G (see below). Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Permutation Group P3 e e a b c d f
e a b c d f
a a b e d f c
b b e a f c d
c c f d e b a
d d c f a e b
f f d c b a e
invariant subgroup U = {e, a, b} ⇒ one coset cU = dU = f U = {c, d, f } factor group P3 /U = {U, cU} U cU U U cU cU cU U
I
We can think of factor groups G/U as coarse-grained versions of G.
I
Often, factor groups G/U are a helpful intermediate step when working out the structure of more complicated groups G.
I
Thus: invariant subgroups are “more useful” subgroups than other subgroups. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Mappings of Groups 0
I
Let G and G be two groups. A mapping φ : G → G 0 assigns to each g ∈ G an element g 0 = φ(g ) ∈ G 0 , with every g 0 ∈ G 0 being the image of at least one g ∈ G.
I
If φ(g1 ) φ(g2 ) = φ(g1 g2 ) ∀g1 , g2 ∈ G, then φ is a homomorphic mapping of G on G 0 . • A homomorphic mapping is consistent with the group structures • A homomorphic mapping G → G 0 is always n-to-one (n ≥ 1):
The preimage of the unit element of G 0 is an invariant subgroup U of G. G 0 is isomorphic to the factor group G/U. I
If the mapping φ is one-to-one, then it is an isomomorphic mapping of G on G 0 . • Short-hand: G isomorphic to G 0 ⇒ G ' G 0 • Isomorphic groups have the same group structure.
I
Examples: • trivial homomorphism
G = P3 and G 0 = {e}
• isomorphism between permutation group P3 and symmetry group C3v of
equilateral triangle
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Products of Groups I
Given two groups G1 = {ai } and G2 = {bk }, their outer direct product is the group G1 × G2 with elements (ai , bk ) and multiplication (ai , bk ) · (aj , bl ) = (ai aj , bk bl ) ∈ G1 × G2 • Check that the group axioms are satisfied for G1 × G2 . • Order of Gn is hn (n = 1, 2) ⇒ order of G1 × G2 is h1 h2 • If G = G1 × G2 , then both G1 and G2 are invariant subgroups of G.
Then we have isomorphisms G2 ' G/G1 and G1 ' G/G2 . • Application: built more complex groups out of simpler groups I
If G1 = G2 = G = {ai }, the elements (ai , ai ) ∈ G ⊗ G define a group G˜ ≡ G ⊗ G called the inner product of G. • The inner product G ⊗ G is isomorphic to G (⇒ same order as G) • Compare: product representations (discussed below)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Matrix Representations of a Group Motivation e = identity i = inversion
I
Consider symmetry group Ci = {e, i}
I
two “types” of basis functions: even and odd more abstract: reducible and irreducible representations
I
Ci e i e e i i i e
matrix representation (based on 1 × 1 and 2 × 2 matrices) Γ1 = {De = 1, Di = 1} consistent with group Γ2 = {De = 1, Di = −1} multiplication table 10 1 0 Γ3 = De = 0 1 , Di = 0 −1 where Γ1 : even function fe (x) = fe (−x) irreducible representations Γ2 : odd functions fo (x) = −fo (−x) Γ3 : reducible representation: decompose any f (x) into even and odd parts fe (x) = 21 f (x) + f (−x) f (x) = fe (x)+fo (x) with fo (x) = 21 f (x) − f (−x) How to generalize these ideas for arbitrary groups? Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Matrix Representations of a Group group G = {gi : i = 1, . . . , h}
I
Let
I
Associate with each gi ∈ G a nonsingular square matrix D(gi ). If the resulting set {D(gi ) : i = 1, . . . , h} is homomorphic to G it is called a matrix representation of G. • gi gj = gk ⇒ D(gi ) D(gj ) = D(gk ) • D(e) = •
I
D(gi−1 )
1 (identity matrix) = D−1 (gi )
dimension of representation = dimension of representation matrices
Example (1): G = C∞ = rotations around a fixed axis (angle φ) I C∞ is isomorphic to group of orthogonal 2 × 2 matrices SO(2) cos φ − sin φ D2 (φ) = ⇒ two-dimensional (2D) representation sin φ cos φ I I I
C∞ is homomorphic to group {D1 (φ) = 1} ⇒ trivial 1D representation 1 0 higher-dimensional C∞ is isomorphic to group ⇒ representation 0 D2 (φ) Generally: given matrix representations of dimensions n1 and n2 , we can construct (n1 + n2 ) dimensional representations Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Matrix Representations of a Group (cont’d)
identity
rotation φ = 120◦
rotation φ = 240◦
reflection y ↔ −y
rotoreflection φ = 120◦
rotoreflection φ = 240◦
Example (2): Symmetry group C3v of equilateral triangle (isomorphic to permutation group P3 )
P3 Koster
e E
a C3
b = a2 C32
c = ec σv
d = ac σv
f = bc σv
Γ1 Γ2
(1) (1)
(1) (1)
(1) (1)
(1) (−1)
(1) (−1)
Γ3
1 0 0 1
(1) (−1)
y x
P3 e multipli- a cation b table c d f
e e a b c d f
a a b e d f c
b b e a f c d
√
− 12 − 23 √ 3 − 12 2
c c f d e b a
d d c f a e b
f f d c b a e
− 21
√
√
3
2
− 23 − 12
1 0 0 −1
√
− 21
− 23
− 23
1 2
√
− 12 √
√
3
2
3
2 1 2
I
mapping G → {D(gi )} homomorphic, but in general not isomorphic (not faithful)
I
consistent with group multiplication table
I
Goal: characterize matrix representations of G
I
Will see: G fully characterized by its “distinct” matrix representations (only three for G = C3v !) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Goal: Identify and Classify Representations I
Theorem 2: If U is an invariant subgroup of G, then every representation of the factor group F = G/U is likewise a representation of G. Proof: G is homomorphic to F, which is homomorphic to the representations of F.
Thus: To identify the representations of G it helps to identify the representations of F. I
Definition: Equivalent Representations Let {D(gi )} be a matrix representation for G with dimension n. Let X be a n-dimensional nonsingular matrix. The set {D0 (gi ) = X D(gi ) X −1 } forms a matrix representation called equivalent to {D(gi )}. Convince yourself: {D0 (gi )} is, indeed, another matrix representation.
Matrix representations are most convenient if matrices {D} are unitary. Thus I
Theorem 3: Every matrix representation {D(gi )} is equivalent to a unitary representation {D0 (gi )} where D0 † (gi ) = D0 −1 (gi )
I
In the following, it is always assumed that matrix representations are unitary. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Theorem 3 (cf. Falicov) Challenge: Matrix X has to be choosen such that it makes all matrices D0 (gi ) unitary simultaneously. I
I
Let {D(gi ) ≡ Di : i = 1, . . . , h} be a matrix representation for G (dimension h). h P Di Di† (Hermitean) Define H = i=1
I
Thus H can be diagonalized by means of a unitary matrix U. P −1 P −1 d ≡ U −1 HU = U Di Di−1 U = U Di U U −1 Di−1 U {z } i i | {z } | P ˜i ˜† =D † =D ˜ ˜ i Di D with dµν = dµ δµν diagonal = i
I
Diagonal entries dµ are positive: PP P P ˜ i )µλ (D ˜ † )λµ = ˜ i )µλ (D ˜ ∗ )µλ = ˜ i )µλ |2 > 0 dµ = (D (D |(D i i i
I I
i
iλ
λ
iλ
±1/2 Take diagonal matrix d˜± with elements (d˜± )µν ≡ dµ δµν P ˜i D ˜ † d˜− (identity matrix) Thus 1 = d˜− d d˜− = d˜− D i i
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Theorem 3 (cont’d) I
˜ i d˜+ = d˜− U −1 Di U d˜+ are unitary matrices Assertion: Di0 = d˜− D equivalent to Di ˜− U −1 • equivalent by construction: X = d • unitarity:
Di0
Di0 †
=1 }| { z P ˜ i d˜+ (d˜− D ˜k D ˜ † d˜− ) d˜+ D ˜ † d˜− = d˜− D k i k P † † ˜i D ˜k D ˜ D ˜ d˜− = d˜− D i k k | {z } | {z } † ˜j = D ˜ =D (rearrangement lemma) j | {z } =d = 1
qed
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Reducible and Irreducible Representations (RRs and IRs) I
If for a given representation {D(gi ) : i = 1, . . . , h}, an equivalent representation {D0 (gi ) : i = 1, . . . , h} can be found that is block diagonal 0 D1 (gi ) 0 0 ∀gi ∈ G D (gi ) = 0 D20 (gi ) then {D(gi ) : i = 1, . . . , h} is called reducible, otherwise irreducible.
I
Crucial: the same block diagonal form is obtained for all representation matrices D(gi ) simultaneously.
I
Block-diagonal matrices do not mix, i.e., if D0 (g1 ) and D0 (g2 ) are block diagonal, then D0 (g3 ) = D0 (g1 ) D0 (g2 ) is likewise block diagonal. ⇒ Decomposition of RRs into IRs decomposes the problem into the smallest subproblems possible.
I
Goal of Representation Theory Identify and characterize the IRs of a group.
I
We will show The number of inequivalent IRs equals the number of classes. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Schur’s First Lemma Schur’s First Lemma: Suppose a matrix M commutes with all matrices D(gi ) of an irreducible representation of G D(gi ) M = M D(gi )
∀gi ∈ G
(♠)
then M is a multiple of the identity matrix M = c 1,
c ∈ C.
Corollaries I
If (♠) holds with M 6= c 1, c ∈ C, then {D(gi )} is reducible.
I
All IRs of Abelian groups are one-dimensional Proof: Take gj ∈ G arbitrary, but fixed. G Abelian ⇒ D(gi ) D(gj ) = D(gj ) D(gi ) ∀gi ∈ G Lemma ⇒ D(gj ) = cj 1 with cj ∈ C, i.e., {D(gj ) = cj } is an IR.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Schur’s First Lemma (cf. Bir & Pikus) I
Take Hermitean conjugate of (♠): M † D† (gi ) = D† (gi ) M † Multiply with D† (gi ) = D−1 (gi ):
I
D(gi ) M † = M † D(gi )
Thus: (♠) holds for M and M † , and also the Hermitean matrices M 0 = 21 (M + M † )
M 00 = 2i (M − M † )
I
It exists a unitary matrix U that diagonalizes M 0 (similar for M 00 ) d = U −1 M 0 U with dµν = dµ δµν
I
Thus (♠) implies D0 (gi ) d = d D0 (gi ), where D0 (gi ) = U −1 D(gi ) U 0 more explicitly: Dµν (gi ) (dµ − dν ) = 0 ∀i, µ, ν
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Schur’s First Lemma (cont’d) Two possibilities: I
All dµ are equal, i.e, d = c 1. So M 0 = UdU −1 and M 00 are likewise proportional to 1, and so is M = M 0 − iM 00 .
I
Some dµ are different: Say {dκ : κ = 1, . . . , r } are different from {dλ : λ = r + 1, . . . , h}. Thus:
0 Dκλ (gi ) = 0
∀κ = 1, . . . , r ; ∀λ = r + 1, . . . , h
Thus {D0 (gi ) : i = 1, . . . , h} is block-diagonal, contrary to the assumption that {D(gi )} is irreducible
qed
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Schur’s Second Lemma Schur’s Second Lemma: Suppose we have two IRs {D1 (gi ), dimension n1 } and {D2 (gi ), dimension n2 }, as well as a n1 × n2 matrix M such that D1 (gi ) M = M D2 (gi )
∀gi ∈ G
(♣)
(1) If {D1 (gi )} and {D2 (gi )} are inequivalent, then M = 0. (2) If M 6= 0 then {D1 (gi )} and {D2 (gi )} are equivalent.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Schur’s Second Lemma (cf. Bir & Pikus) I
Take Hermitean conjugate of (♣); use D† (gi ) = D−1 (gi ) = D(gi−1 ), so M † D1 (gi−1 ) = D2 (gi−1 )M †
I
Multiply by M on the left; Eq. (♣) implies M D2 (gi−1 ) = D1 (gi−1 ) M, so MM † D1 (gi−1 ) = D1 (gi−1 )MM † ∀gi−1 ∈ G
I
Schur’s first lemma implies that MM † is square matrix with with c ∈ C MM † = c 1
I
(*)
Case a: n1 = n2 • If c 6= 0 then det M 6= 0 because of (*), i.e., M is invertible.
So (♣) implies
M −1 D1 (gi ) M = D2 (gi ) thus {D1 (gi )} and {D2 (gi )} are equivalent.
∀gi ∈ G
• If c = 0 then MM † = 0, i.e.,
P
† Mµν Mνµ =
ν
P ν
∗ = Mµν Mµν
P
|Mµν |2 = 0
∀µ
ν
so that M = 0. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Schur’s Second Lemma (cont’d) I
Case b: n1 6= n2
(n1 < n2 to be specific)
˜ with det M ˜ = 0. • Fill up M with n2 − n1 rows to get matrix M ˜M ˜ † = MM † , so that • However M ˜M ˜ † ) = (det M) ˜ (det M ˜ †) = 0 det(MM † ) = det(M • So c = 0, i.e., MM † = 0, and as before M = 0.
qed
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Orthogonality Relations for IRs Notation: I Irreducible Representations (IR): ΓI = {DI (gi ) : gi ∈ G} I nI = dimensionality of IR ΓI I h = order of group G
Theorem 4: Orthogonality Relations for Irreducible Representations (1) two inequivalent IRs ΓI 6= ΓJ h P ∀ µ0 , ν 0 = 1, . . . , nI DI (gi )∗µ0 ν 0 DJ (gi )µν = 0 ∀ µ, ν = 1, . . . , nJ i=1 (2) representation matrices of one IR ΓI h nI P DI (gi )∗µ0 ν 0 DI (gi )µν = δµ0 µ δν 0 ν h i=1
∀ µ0 , ν 0 , µ, ν = 1, . . . , nI
Remarks I [DI (gi )µν : i = 1, . . . , h] form vectors in a h-dim. vector space p I vectors are normalized to h/nI (because ΓI assumed to be unitary) I I
vectors for different I , µν are orthogonal P P in total, we have I nI2 such vectors; therefore I nI2 ≤ h
Corollary: For finite groups the number of inequivalent IRs is finite. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Theorem 4: Orthogonality Relations for IRs (1) two inequivalent IRs ΓI 6= ΓJ I Take arbitrary nJ × nI matrix X 6= 0 P I Let M ≡ DJ (gi ) X DI (gi−1 ) i
=M
(i.e., at least one Xµν 6= 0) =1
z }| {z }| { DJ (gk ) DJ (gi ) X DI (gi−1 ) DI−1 (gk ) DI (gk ) ⇒ DJ (gk ) M = i | {z } | {z } P DJ (gk gi ) X DI−1 (gk gi ) DI (gk ) = |{z} |{z} i =gj =gj P −1 = DJ (gj ) X DI (gj ) DI (gk ) P
j
|
{z } = M DI (gk ) ⇒ (Schur’s Second Lemma) 0 = Mµµ0 ∀µ, µ0 PP in particular correct for DJ (gi )µκ Xκλ DI (gi−1 )λµ0 = Xκλ = δνκ δλν 0 i κ,λ P −1 = DJ (gi )µν DI (gi )ν 0 µ0 i P = DI (gi )∗µ0 ν 0 DJ (gi )µν qed i
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Theorem 4: Orthogonality Relations for IRs (cont’d) (2) representation matrices of one IR ΓI First steps similar to case (1): P I Let M ≡ DI (gi ) X DI (gi−1 ) with nI × nI matrix X 6= 0 i
⇒ DI (gk ) M = M DI (gk ) ⇒ (Schur’s First Lemma): M = c 1, c ∈ C PP I Thus c δµµ0 = DI (gi )µκ Xκλ DI (gi−1 )λµ0 i κ,λ P = DI (gi )µν DI (gi−1 )ν 0 µ0 = Mµµ0
choose Xκλ = δνκ δλν 0
i
I
1P P h 1P c= Mµµ = DI (gi )µν DI (gi−1 )ν 0 µ = δνν 0 nI µ nI i µ nI | {z } DI (gi−1 gi = e)ν 0 ν = δνν 0
qed
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Goal: Characterize different irreducible representations of a group
Characters I
The traces of the representation matrices are called characters P χ(gi ) ≡ tr D(gi ) = i D(gi )µµ
I
Equivalent IRs are related via a similarity transformation D0 (gi ) = X D(gi )X −1
with X nonsingular
This transformation leaves the trace invariant: tr D0 (gi ) = tr D(gi ) ⇒ Equivalent representations have the same characters. I
Theorem 5: If gi , gj ∈ G belong to the same class Ck of G, then for every representation ΓI of G we have χI (gi ) = χI (gj ) Proof: • gi , gj ∈ C ⇒ ∃ x ∈ G
with
gi = x gj x −1
• Thus DI (gi ) = DI (x) DI (gj ) DI (x −1 ) • χI (gi ) = tr DI (x) DI (gj ) DI (x −1 )
(trace invariant under cyclic permutation)
= tr DI (x −1 ) DI (x) DI (gj ) = χI (gk ) | {z } =1 Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Characters (cont’d) Notation I χI (Ck ) denotes the character of group elements in class Ck I
The array [χI (Ck )] with I = 1, . . . , N ˜ k = 1, . . . , N is called character table.
(N = number of IRs) ˜ = number of classes) (N
Remark: For Abelian groups the character table is the table of the 1 × 1 representation matrices
Theorem 6: Orthogonality relations for characters Let {DI (gi )} and {DJ (gi )} be two IRs of G. Let hk be the number of ˜ the number of classes. Then elements in class Ck and N ˜ N h P k ∗ Proof: Use orthogonality χI (Ck ) χJ (Ck ) = δIJ ∀ I , J = 1, . . . , N relation for IRs k=1 h ˜ of character table I Interpretation: rows [χI (Ck ) : k = 1, . . . N] ˜ are like N orthonormal vectors in a N-dimensional vector space ˜ ⇒ N ≤ N. I
If two IRs ΓI and ΓJ have the same characters, this is necessary and sufficient for ΓI and ΓJ to be equivalent. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Symmetry group C3v of equilateral triangle
identity
rotation φ = 120◦
rotation φ = 240◦
reflection y ↔ −y
rotoreflection φ = 120◦
rotoreflection φ = 240◦
(isomorphic to permutation group P3 )
P3 Koster
e E
a C3
b = a2 C32
c = ec σv
d = ac σv
f = bc σv
Γ1 Γ2
(1) (1)
(1) (1)
(1) (1)
(1) (−1)
(1) (−1)
Γ3
1 0 0 1
(1) (−1)
y x
P3 e multipli- a cation b table c d f
e e a b c d f
a a b e d f c
√
− 12 − 23 √ 3 − 12 2
b b e a f c d
c c f d e b a
d d c f a e b
− 21
√
√
3
2
− 23 − 12
f f d c b a e
1 0 0 −1
√
− 21
− 23
− 23
1 2
√
− 12 √ 2
3
√
3
2 1 2
Character table P3 e a, b c, d, f C3v E 2C3 3σv Γ1 1 1 1 Γ2 1 1 −1 Γ3 2 −1 0 Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Interpretation: Character Tables I
A character table is the uniquely defined signature of a group and its IRs ΓI [independent of, e.g., phase conventions for representation matrices DI (gi ) that are quite arbitrary].
I
Isomorphic groups have the same character tables.
I
Yet: the labeling of IRs ΓI is a matter of convention. – Customary: • Γ1 = identity representation: all characters are 1 • IRs are often numbered such that low-dimensional IRs come first;
higher-dimensional IRs come later • If G contains the inversion, a superscript ± is added to ΓI indicating
the behavior of Γ± I under inversion (even or odd) • other labeling schemes are inspired by compatibility relations
(more later) I
Different authors use different conventions to label IRs. To compare such notations we need to compare the uniquely defined characters for each class of an IR. (See, e.g., Table 2.7 in Yu and Cardona: Fundamentals of Semiconductors; here we follow Koster et al.) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Decomposing Reducible Representations (RRs) Into Irreducible Representations (IRs) Given an arbitrary RR {D(gi )} the representation matrices {D(gi )} can be brought into block-diagonal form by a suitable unitary transformation
D(gi )
→
D (g ) 1 i .. . D1 (gi ) .. D0 (gi ) = . DN (gi ) ..
0
0
. DN (gi )
) a1 times .. . ) aN times
Theorem 7: Let aI be the multiplicity, with which the IR ΓI ≡ {DI (gi )} is contained in the representation {D(gi )}. Then N P (1) χ(gi ) = aI χI (gi ) I =1 ˜ h N P 1P hk ∗ (2) aI = χ∗I (gi ) χ(gi ) = χI (Ck ) χ(Ck ) h i=1 k=1 h
We say: {D(gi )} contains the IR ΓI aI times. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof: Theorem 7 (1) due to invariance of trace under similarity transformations N h X 1X (2) we have aJ χJ (gi ) = χ(gi ) χ∗I (gi ) × h i=1
J=1
⇒
N X J=1
h h 1X ∗ 1X ∗ aJ χI (gi ) χJ (gi ) = χI (gi ) χ(gi ) h h i=1 i=1 | {z }
qed
=δIJ
Applications of Theorem 7: I
Corollary: The representation {D(gi )} is irreducible if and only if h X |χ(gi )|2 = h i=1
Proof: Use Theorem 7 with aI = I
1 0
for one I otherwise
Decomposition of Product Representations (see later) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Where Are We? We have discussed the orthogonality relations for I
irreducible representations
I
characters
These can be complemented by matching completeness relations. Proving those is a bit more cumbersome. It requires the introduction of the regular representation.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
The Regular Representation Finding the IRs of a group can be tricky. Yet for finite groups we can derive the regular representation which contains all IRs of the group. I
Interpret group elements gν as basis vectors {|gν i : ν = 1, . . . h} for a h-dim. representation
⇒ Regular representation: νth column vector of DR (gi ) gives image |gµ i = gi |gν i ≡ |gi gν i of basis vector |gν i 1 if gµ gν−1 = gi ⇒ DR (gi )µν = 0 otherwise I
Strategy: • Re-arrange the group multiplication table
as shown on the right • For each gi ∈ G we have DR (gi )µν = 1,
if the entry (µ, ν) in the re-arranged group multiplication table equals gi , otherwise DR (gi )µν = 0.
g1 g2 g3 .. .
g1−1 g2−1 g3−1 . . . e ... e .. e . e
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Properties of the Regular Representation {DR (gi )} (1) {DR (gi )} is, indeed, a representation for the group G (2) It is a faithful representation, i.e., {DR (gi )} is isomorphic to G = {gi }. h if gi = e (3) χR (gi ) = 0 otherwise Proof: (1) Matrices {DR (gi )} are nonsingular, as every row / every column contains “1” exactly once. Show: if gi gj = gk , then DR (gi )DR (gj ) = DR (gk ) Take i, j, µ, ν arbitrary, but fixed DR (gi )µλ = 1 only for gµ gλ−1 = gi ⇔ gλ = gi−1 gµ DR (gj )λν = 1 only for gλ gν−1 = gj ⇔ gλ = gj gν P ⇔ DR (gi )µλ DR (gj )λν = 1 only for gi−1 gµ = gj gν λ
⇔ gµ gν−1 = gi gj = gk
[definition of DR (gk )µν ]
(2) immediate consequence of definition of DR (gi ) 1 if gi = gµ gµ−1 = e (3) DR (gi )µµ = 0 otherwise P h if gi = e ⇒ χR (gi ) = DR (gi )µµ = 0 otherwise µ Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Regular Representation for P3 g e a b c d f
e e a b c d f
a a b e d f c
b b e a f c d
c c f d e b a
d d c f a e b
Thus
f f d c b a e
⇒
g −1 e a b c d f
e e a b c d f
b b e a f c d
a a b e d f c
c c f d e b a
d d c f a e b
f f d c b a e
1 0 0 0 1 0 0 0 1 e= 1 0 0 0 1 0 0 0 1
0 0 1 1 0 0 0 1 0 a= 0 1 0 0 0 1 1 0 0
0 1 0 0 0 1 1 0 0 b= 0 0 1 1 0 0 0 1 0
1 0 0 0 1 0 0 0 1 c= 1 0 0 0 1 0 0 0 1
0 1 0 0 0 1 1 0 0 d = 0 0 1 1 0 0 0 1 0
0 0 1 1 0 0 0 1 0 f = 0 1 0 0 0 1 1 0 0
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Completeness of Irreducible Representations Lemma: The regular representation contains every IR nI times, where nI = dimensionality of IR ΓI . P Proof: Use Theorem 7: χR (gi ) = aI χI (gi ) where i P ∗ ∗ 1 1 aI = h χI (gi ) χR (gi ) = h χI (e) χR (e) = nI i | {z } | {z } =nI
=h
Corollary (Burnside’s Theorem): For a group G of order h, the dimensionalities nI of the IRs ΓI obey P P P 2 Proof: h = χR (e) = I aI χI (e) = I nI2 nI = h I
serious constraint for dimensionalities of IRs
Theorem 8: The representation matrices DI (gi ) of a group G of order h obey the completeness relation X X nI DI∗ (gi )µν DI (gj )µν = δij ∀ i, j = 1, . . . , h (*) h µ,ν I
Proof: I Theorem 4: Interpret [DI (gi )µν : i = 1, . . . , h] as orthonormal row vectors of a matrix M ⇒ M is square matrix: unitary ⇒ column vectors also orthonormal I Corollary: M has h columns = completeness (*) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Completeness Relation for Characters Theorem 9: Completeness Relation for Characters If χI (Ck ) is the character for class Ck and irreducible representation I , then hk X ∗ ˜ χI (Ck )χI (Ck 0 ) = δkk 0 ∀ k, k 0 = 1, . . . , N h I
χ1 (Ck ) .. . χN (Ck )
I
Interpretation:
columns
˜ of character table [k = 1, . . . , N]
˜ orthonormal vectors in a N-dimensional vector space are like N I
˜ ≤ N. Thus N
(from completeness)
I
˜ Also N ≤ N
(from orthogonality)
I
Character table
)
Number N of irreducible representations ˜ of classes = Number N
• square table • rows and column form orthogonal vectors Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Theorem 9: Completeness Relation for Characters Lemma: Let {DI (gi )} be an nI -dimensional IR of G. Let Ck be a class of G with hk elements. Then X hk χI (Ck ) 1 DI (gi ) = nI i∈Ck
The sum over all representation matrices in a class of an IR is proportional to the identity matrix.
Proof of Lemma: I
For arbitrary P gj ∈ G P P DI (gi 0 ) DI (gj ) DI (gi ) DI (gj−1 ) = DI (gj ) DI (gi ) DI (gj−1 ) = x {z } i 0 ∈Ck i∈Ck i∈Ck | =DI (gi 0 ) with i 0 ∈Ck
because gj maps gi1 6= gi2 onto gi10 6= gi20
⇒ (Schur’s First Lemma):
P i∈Ck
I
ck =
DI (gi ) = ck 1
h 1 P k tr DI (gi ) = χI (Ck ) nI n I i∈Ck
qed
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Proof of Theorem 9: Completeness Relation for Characters I
Use Theorem 8 (Completeness Relations N P n P I DI∗ (gi )µν DI (gj )µν = δij h I =1 µ,ν
for Irreducible Representations)
P P
i∈Ck j∈Ck 0
P P nI P P ∗ ⇒ DI (gi ) DI (gj ) = hk δkk 0 j∈Ck 0 I h µ,ν i∈Ck µν µν | {z }| {z } hk ∗ hk 0 χ (Ck ) δµν χI (Ck 0 ) δµν (Lemma) nI I nI {z } | P hk hk 0 ∗ χI (Ck ) χI (Ck 0 ) δµν nI2 |µ,ν{z }
qed
=nI
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Summary: Orthogonality and Completeness Relations Theorem 4: Orthogonality Relations for Irreducible Representations h I , J = 1, . . . , N nI X µ0 , ν 0 = 1, . . . , nI DI (gi )∗µ0 ν 0 DJ (gi )µν = δIJ δµµ0 δνν 0 h µ, ν = 1, . . . , nJ i=1
Theorem 8: Completeness Relations for Irreducible Representations N X X nI I =1 µ,ν
h
DI∗ (gi )µν DI (gj )µν = δij
∀ i, j = 1, . . . , h
Theorem 6: Orthogonality Relations for Characters ˜ N X hk k=1
h
χ∗I (Ck ) χJ (Ck ) = δIJ
∀ I , J = 1, . . . , N
Theorem 9: Completeness Relation for Characters N hk X ∗ χI (Ck )χI (Ck 0 ) = δkk 0 h
˜ ∀ k, k 0 = 1, . . . , N
I =1
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Unreducible hhh hhhh
More on Irreducible h h Problems
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Quantum Mechanics Topics: I
Behavior of quantum mechanical states and operators under symmetry operations
I
Relation between irreducible representations and invariant subspaces of the Hilbert space
I
Connection between eigenvalue spectrum of quantum mechanical operators and irreducible representations
I
Selection rules: symmetry-induced vanishing of matrix elements and Wigner-Eckart theorem
Note: Operator formalism of QM convenient to discuss group theory. Yet: many results also applicable in other areas of physics. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Symmetry Operations in Quantum Mechanics (QM) I
Let G = {gi } be a group of symmetry operations of a qm system e.g., translations, rotations, permutation of particles
I
Translated into the language of group theory: In the Hilbert space of the qm system we have a group of unitary ˆ i )} such that G 0 is isomorphic to G. operators G 0 = {P(g
Examples I
translations Ta ˆ a ) = exp (i p ˆ · a/~) → unitary operator P(T (ˆ p = momentum) 2 1 ˆ P(Ta ) ψ(r) = 1 + ∇ · a + (∇ · a) + . . . ψ(r) = ψ(r + a) 2
I
rotations Rφ (Lˆ = angular momentum ˆ ˆ · nφ/~ → unitary operator P(n, φ) = exp i L φ = angle of rotation n = axis of rotation)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Transformation of QM States I I
I
Let {|νi} be an orthonormal basis ˆ i ) be the symmetry operator for the symmetry transformation gi Let P(g with symmetry group G = {gi }. ˆ i ) |νi = P |µi hµ| P(g ˆ i ) |νi Then P(g | {z } ↑ µ 1=
P
I
So
µ
|µihµ|
D(gi )µν
matrix of a unitary
ˆ i ) |νi = P D(gi )µν |µi P(g
where D(gi )µν = representation of G
ˆ i ) unitary because P(g
µ I
Note: bras and kets transform according to complex conjugate representations ˆ i )† = P hµ| D(gi )∗µν hν|P(g µ
I
Let gi , gj ∈ G with gi gj = gk ∈ G. Then D(gi )κµ
D(gi )µν
(consistent with matrix multiplication)
}| {z }| { P z ˆ i ) |µi hµ| P(g ˆ i ) |νi = P D(gi )κµ D(gi )µν |κi |κihκ| P(g κµ κµ ˆ i ) P(g ˆ j ) |νi = P(g P P(g ˆ k ) |νi = D(gk )κν |κi κ
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Transformation of (Wave) Functions ψ(r) I
ˆ i )|ri = |r0 =gi ri ⇔ hr|P(g ˆ i ) = hr0 =g −1 r| P(g i
I
Let ψ(r) ≡ hr|ψi
ˆ )† = P(g ˆ −1 ) b/c P(g
ˆ i ) ψ(r) ≡ hr|P(g ˆ i )|ψi = ψ(g −1 r) ≡ ψi (r) ⇒ P(g i I
In general, the functions V = {ψi (r) : i = 1, . . . , h} are linear dependent ⇒ Choose instead linear independent functions ψν (r) ≡ hr|νi spanning V ˆ i ) ψν (r) in terms of {ψν (r)}: ⇒ Expand images P(g ˆ i ) ψ(r) = hr|P(g ˆ i )|ψi = Phr|µihµ|P(g ˆ i )|νi = PD(gi )µν ψµ (r) P(g µ
µ
ˆ i ) ψν (r) = ψν (g −1 r) = P D(gi )µν ψµ (r) ⇒ P(g i µ
I
Thus: every function ψ(r) induces a matrix representation Γ = {D(gi )}
I
Also: every representation Γ = {D(gi )} is completely characterized by a (nonunique) set of basis functions {ψν (r)} transforming according to Γ.
I
Dirac bra-ket notation convenient for formulating group theory of functions. Yet: results applicable in many areas of physics beyond QM. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Important Representations in Physics (usually reducible) (1) Representations for polar and axial (cartesian) vectors I
generally: two types of point group symmetry operations • proper rotations gpr = (n, θ) about axis n, angle θ
D[gpr = (n, θ)] =
Rodrigues’ rotation formula
nx2 (1 − cos θ) + cos θ ny nx (1 − cos θ) + nz sin θ
nx ny (1 − cos θ) − nz sin θ nx nz (1 − cos θ) + ny sin θ 2 ny (1 − cos θ) + cos θ ny nz (1 − cos θ) − nx sin θ nz nx (1 − cos θ) − ny sin θ nz ny (1 − cos θ) + nx sin θ nz2 (1 − cos θ) + cos θ I
det D(gpr ) = +1
I
χ(gpr ) = tr D(gpr ) = 1 + 2 cos θ
independent of n
• improper rotations gim ≡ i gpr = gpr i where i = inversion I
polar vectors • proper rotations gpr : I I
det Dpol (gpr ) = +1 tr Dpol (gpr ) = 1 + 2 cos θ
• inversion i: Dpol (i) = −13×3 • improper rotations gim = i gpr : I D pol (gim ) = −Dpol (gpr ) I det D pol (gim ) = −1 I tr D pol (gim ) = −(1 + 2 cos θ)
• Γpol = {Dpol (g )} ⊆ O(3) always a faithful representation (i.e., isomorphic to G) • examples: position r, linear momentum p, electric field E Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Important Representations in Physics (1) Representations for polar and axial (cartesian) vectors (cont’d) I
axial vectors • proper rotations gpr : I
Dax (gpr ) = Dpol (gpr )
I
det Dax (gpr ) = +1
I
tr Dax (gpr ) = 1 + 2 cos θ
• inversion i: Dax (i) = +13×3 • improper rotations gim = i gpr : I
Dax (gim ) = Dax (gpr ) = −Dpol (gpr )
I
det Dax (gim ) = +1
I
tr Dax (gim ) = 1 + 2 cos θ
• Γax = {Dax (g )} ⊆ SO(3) • examples: angular momentum L, magnetic field B I
systems with discrete symmetry group G = {gi : i = 1, . . . , h}: Γpol = {Dpol (gi ) : i = 1, . . . , h} Γax = {Dax (gi ) : i = 1, . . . , h}
We have a “universal recipe” to construct the 3 × 3 matrices Dpol (g ) and Dax (g ) for each group element gpr = (n, θ) and gim = i (n, θ) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Important Representations in Physics (cont’d) (2) Equivalence Representations Γeq I Consider symmetric object (symmetry group G) • vertices, edges, and faces of platonic solids are equivalent by symmetry • atoms / atomic orbitals |µi in a molecule may be equivalent by symmetry I I
Equivalence representation Γeq describes mapping of equivalent objects ˆ ) |µi = P Deq (g )νµ |νi Generally: P(g ν
I
Example: orbitals of equivalent H atoms in NH3 molecule (group C3v ) Equivalent to: permutations of corners of triangle (group P3 )
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Symmetry group C3v of equilateral triangle (isomorphic to permutation group P3 ) identity
rotation φ = 120◦
rotation φ = 240◦
reflection y ↔ −y
rotoreflection φ = 120◦
rotoreflection φ = 240◦
z
P3 Koster
e E
a C3
b = a2 C32
c = ec σv
d = ac σv
f = bc σv
Γ1 Γ2
(1) (1)
(1) (1)
(1) (1)
(1) (−1)
(1) (−1)
Γ3
1 0 0 1
(1) (−1)
y 2
3 1
x
n, θ Γpol = Γ1 + Γ 3 Γax = Γ2 + Γ 3
√
− 12 − 23 √ 3 − 12 2
1 1 1
1 1 1
Γeq = 1 1 Γ1 + Γ 3 1
− 12 √ 3 2 − 12 √ 3 2
√ − 23 − 21 √ − 23 − 21
√
− 12
√ − 23
(0, 0, 1), 0 (0, 0, 1), 2π/3
!
− 12
√ 3 2 3 − 12 2
−
√ 3 2 3 1 − 2 2
1
−1 1
! −1
− 12 √ −
(0, 1, 0), π
1
1 −1
1
1
1
1
1 1
1
√ − 23 √ 1 − 23 2 √ 3 1 (− 2 , − 2 , 0), √ − 12 − 23 √ 1 − 23 2
1 0 0 −1
! 1
− 12 √
1
(0, 0, 1), 4π/3
1
!
3
2
− 21
1 2 √
√ 3
√
π
!
(−
3
2
3
1 1 1
, 21 , 0), π
− 21 √ 3 2
√
!
3
2 1 2 √
1
!
− 23 − 23 − 12 1 2 √
−1
3
2 1 2
2 √
!
− 21
1 1
√
− 12
1 3
2
2
−1
1 1 1
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Transformation of QM States (cont’d) I
in general: representation {D(gi )} of states {|νi} is reducible
I
We have D0 (gi ) = U −1 D(gi ) U P −1 More explicitly: D0 (gi )µ0 ν 0 = Uµ0 µ D(gi )µν Uνν 0 | {z } µν
I
ˆ i ) |νi hµ| P(g
=
P µν
ˆ hµ|Uµ−1 0 µ P(gi ) Uνν 0 |νi
ˆ i ) |ν 0 i = hµ0 | P(g 0
P with |ν i = Uνν 0 |νi ν I
Thus: block diagonalization {D(gi )} → {D0 (gi ) = U −1 D(gi ) U} corresponds to change of basis P {|νi} → {|ν 0 i = Uνν 0 |νi} ν Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Basis Functions for Irreducible Representations I
matrices {DI (gi )} are fully characterized by basis functions {ψνI (r) : ν = 1, . . . nI } transforming according to IR ΓI P ˆ i ) ψνI (r) = ψν (g −1 r) = P(g DI (gi )µν ψµI (r) i µ
I I
convenient if we need to spell out phase conventions for {DI (gi )} (→ Koster) identify IRs for (components of) polar and axial vectors
identity
rotation φ = 120◦
rotation φ = 240◦
reflection y ↔ −y
rotoreflection φ = 120◦
rotoreflection φ = 240◦
z
Γ1
(1)
(1)
(1)
(1)
(1)
(1)
Γ2
(1)
Γ3
1 0 0 1
y
x
1 0 0 1
(1)
√
− 21 − 23 √ 3 − 12 2 √
− 21 − 23 √ 3 − 12 2
(1)
√ 3 2 √ 3 − 2 − 12 √ 3 − 12 2 √ − 23 − 12
− 12
(−1) (−1) 1 √3 1 0 0 −1
2 √
2
3
2
−1 0 0 1
− 12 √
− 21
− 23
− 23
1 2
√
basis functions
Example: Symmetry group C3v
x 2 + y 2 ; z; L2x + L2y Lz
(−1) 1 2 √ − 23
− 12 √ 3 2
√
− 23 − 12 √ 2 1 2
3
x, y
Lx , Ly
r = (x, y , z) = polar vector, L = (Lx , Ly , Lz ) = axial vector Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Relevance of Irreducible Representations
Invariant Subspaces Definition: I Let G = {gi } be a group of symmetry transformations. Let H = {|µi} be a Hilbert space with states |µi.
I
A subspace S ⊂ H is called invariant subspace (with respect to G) if ˆ i ) |µi ∈ S P(g ∀ gi ∈ G, ∀ |µi ∈ S If an invariant subspace can be decomposed into smaller invariant subspaces, it is called reducible, otherwise it is called irreducible.
Theorem 10: An invariant subspace S is irreducible if and only if the states in S transform according to an irreducible representation. Proof:
I
Suppose {D(gi )} is reducible. ∃ unitary transformation U with {D0 (gi ) = U −1 D(gi ) U} block diagonal P For {D0 (gi )} we have the basis {|µ0 i = µ Uµµ0 |µi}
I
The block diagonal form of {D0 (gi )} implies that {|µ0 i is reducible
I I
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Invariant Subspaces (cont’d) Corollary: Every Hilbert space H can be decomposed into irreducible invariant subspaces SI transforming according to the IR ΓI Remark: Given a Hilbert space H we can generally have multiple (possibly orthogonal) irreducible invariant subspaces SIα SIα = |I ν αi : ν = 1, . . . , nI transforming according to the same IR ΓI P ˆ i ) |I ν αi = P(g DI (gi )µν |I µαi µ
Theorem 11: (1) States transforming according to different IRs are orthogonal (2) For states |I µαi and |I ν βi transforming according to the same IR ΓI we have hI µα | I ν βi = δµν hI α||I βi where the reduced matrix element hI α||I βi is independent of µ, ν. Remark: This theorem lets us anticipate the Wigner-Eckart theorem Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Invariant Subspaces (cont’d) Proof of Theorem 11 ˆ j ): I Use unitarity of P(g I
ˆ j )† P(g ˆ j) = 1 = P(g
1 h
P ˆ i )† P(g ˆ i) P(g i
Then P ˆ i )† P(g ˆ i ) |J ν βi hI µα |J ν βi = h1 hI µα| P(g {z } | {z } i | P P µ0
=
P µ0 ν 0
=
hI µ0 α|DI (gi )∗ µ0 µ
hI µ0 α |J ν 0 βi h1 |
δIJ δµν n1I |
P µ0
ν0
P
i
DJ (gi )ν 0 ν |J ν 0 βi
DI (gi )∗µ0 µ DJ (gi )ν 0 ν {z }
(1/nI ) δIJ δµν δµ0 ν 0
0
hI µ α |I µ0 βi {z }
≡hI α||I βi
qed
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Discussion Theorem 11 I
ΓJ × ΓI contains the identity representation Γ1 if and only if the IR ΓJ is the complex conjugate of ΓI , i.e., Γ∗J = ΓI ⇔ DJ (g )∗ = DI (g ) ∀g .
I
If the ket |J µαi transforms according to the IR ΓJ , the bra hJ µα| transforms according to the complex conjugate representation Γ∗J .
I
Thus: hJ µα|I ν βi = 6 0 equivalent to • bra and ket transform according to complex conjugate representations • hJ µα|I ν βi contains the identity representation
I
Indeed, common theme of representation theory applied to physics: Terms are only nonzero if they transform according to a representation that contains the identity representation.
I
Variant of Theorem 11 (Bir & Pikus): If fI (x) transforms according to some IR ΓI , then only if ΓI is the identity representation.
I
Z
fI (x) dx 6= 0
Applications • Wigner-Eckart Theorem • Nonzero elements of material tensors • Our universe would be zero “by symmetry” if the apparently trivial identity
representation did not exist.
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Decomposition into Irreducible Invariant Subspaces I
I I
Goal: Decompose general state |ψi ∈ H into components from irreducible invariant subspaces SI ˆ i) ˆ Iµµ0 := nI P DI (gi )∗µµ0 P(g Generalized projection operator Π h ˆ I 0 |Jναi = δIJ δµ0 ν |I µαi Π µµ I ˆJ 0 ˆ ˆ J 0 = δIJ δµ0 ν Π (ii) Πµµ0 Π νν µν P ˆI Πµµ = 1 (iii)
i
Theorem 12: (i)
Proof: ˆ Iµµ0 |Jναi = (i) Π
Iµ
nI h
P i
ˆ i ) |Jναi = DI (gi )∗µµ0 P(g | {z } P
(ii)
ˆ Iµµ0 Π
ˆ Jνν 0 Π
= = =
ν0
P
nI
ν0
|h
DJ (gi )ν 0 ν |Jν 0 αi
DI (gi )∗µµ0 DJ (gj )∗νν 0
∗ i DI (gi )µµ0
P
{z
DJ (gi )ν 0 ν |Jν 0 αi }
δIJ δµν 0 δµ0 ν
ˆ i ) P(g ˆ j) P(g
nI h
P
nI h
ˆ k) DI (gi )∗µµ0 DJ (gi−1 gk )∗νν 0 P(g P nI P ˆ k) DI (gi )∗µµ0 DJ (gi−1 )∗νλ DJ (gk )∗λν 0 P(g h | {z } i kλ | {z DJ (gi )λν } [or use (i)]
nJ h
i P i
nJ h nJ h
P
subst. gi gj = gk
j P k
δIJ δµλ δµ0 ν
(iii)
P ˆI P1 P P P ∗ ˆ ˆ i ) = P(e) ˆ Πµµ = δgi e P(g ≡1 µ DI (gi )µµ0 nI P(gi ) = h i i Iµ I | {z } |{z} χ∗ (gi ) I
χI (e)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Decomposition into Invariant Subspaces (cont’d) Discussion I
Let |ψi =
P
cJνα |Jναi general state with coefficients cJµα
(*)
Jνα I
ˆ Iµµ projects |ψi on components |I µαi: Diagonal operator Π ˆ Iµµ )2 |ψi = Π ˆ Iµµ |ψi = • (Π
P
cI µα |I µαi
α
P ˆI • Πµµ = 1 Iµ
I
ˆ I ≡ PΠ ˆ Iµµ = Let Π µ
ˆ I |ψi = • Π
P
nI h
P
ˆ i) : χ∗I (gi ) P(g
i
cI να |I ναi
να
ˆI
• Π projects |ψi on the invariant subspace SI (IR ΓI ) I
For functions ψ(r) ≡ hr|ψi: ˆ Iµµ0 ψ(r) = nI P DI (gi )∗µµ0 ψ(g −1 r) Π i h i
we need not know the expansion (*) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Irreducible Invariant Subspaces (cont’d) Example: e = identity i = inversion
I
Group Ci = {e, i}
I
character table
I
ˆ P(e) ψ(x) = ψ(x),
I
ˆI = Projection operator Π
I
Ci e i e e i i i e
Ci e i Γ1 1 1 Γ2 1 −1
ˆ ψ(x) = ψ(−x) P(i) nI h
P
ˆ i) χ∗I (gi ) P(g
with nI = 1, h = 2
i
ˆ1 = Π
1 2
ˆ ˆ [P(e) + P(i)]
ˆ 1 ψ(x) = ⇒ Π
1 2
[ψ(x) + ψ(−x)] even part
ˆ2 = Π
1 2
ˆ ˆ [P(e) − P(i)]
ˆ 2 ψ(x) = ⇒ Π
1 2
[ψ(x) − ψ(−x)] odd part
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Product Representations I
Let {|I µi : µ = 1, . . . nI } and {|Jνi : ν = 1, . . . nJ } denote basis functions for invariant subspaces SI and SJ (need not be irreducible) Consider the product functions {|I µi |Jνi : µ = 1, . . . , nI ; ν = 1, . . . , nJ }. How do these functions transform under G?
I
Definition: Let DI (g ) and DJ (g ) be representation matrices for g ∈ G. The direct product (Kronecker product) DI (g ) ⊗ DJ (g ) denotes the matrix whose elements in row (µν) and column (µ0 ν 0 ) are given by µ, µ0 = 1, . . . , nI ν, ν 0 = 1, . . . , nJ x11 x12 y11 y12 Example: Let DI (g ) = and DJ (g ) = x21 x22 y21 y22 x y x 11 11 11 y12 x12 y11 x12 y12 x11 y21 x11 y22 x12 y21 x12 y22 x11 DJ (g ) x12 DJ (g ) DI (g ) ⊗ DJ (g ) = = x21 y11 x21 y12 x22 y11 x22 y12 x21 DJ (g ) x22 DJ (g ) x21 y21 x21 y22 x22 y21 x22 y22
[DI (g ) ⊗ DJ (g )]µν,µ0 ν 0 = DI (g )µµ0 DJ (g )νν 0
I
I
Details of the arrangement in the following not relevant Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Product Representations (cont’d) I
Dimension of product matrix dim [DI (g ) ⊗ DJ (g )] = dim DI (g ) dim DJ (g )
I
Let ΓI = {DI (gi )} and ΓJ = {DJ (gi )} be representations of G. Then ΓI × ΓJ ≡ {DI (g ) ⊗ DJ (g )} is a representation of G called product representation.
I
ΓI × ΓJ is, indeed, a representation: Let DI (gi ) DI (gj ) = DI (gk ) and DJ (gi ) DJ (gj ) = DJ (gk ) ⇒ [DI (gi ) ⊗ DJ (gi )] [DI (gj ) ⊗ DJ (gj )] µν,µ0 ν 0 P = DI (gi )µκ DJ (gi )νλ DI (gj )κµ0 DJ (gj )λν 0 κλ
→ DI (gk )µµ0 = [DI (gk ) ⊗ DJ (gk )]µν,µ0 ν 0 I
→ DJ (gk )νν 0
P 0 ˆ ) |I µi = 0 Let P(g µ0 DI (g )µ µ |I µ i P 0 ˆ ) |Jνi = 0 P(g ν 0 DJ (g )ν ν |Jν i P ˆ ) |I µi|Jνi = Then P(g [DI (g ) ⊗ DJ (g )]µ0 ν 0 ,µν |I µ0 i|Jν 0 i µ0 ν 0
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Product Representations (cont’d) I
The characters of the product representation are χI ×J (gi ) = χI (gi ) χJ (gi )
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Decomposing Product Representations I
Let ΓI = {DI (gi )} and ΓJ = {DJ (gi )} be irreducible representations of G The product representation ΓI × ΓJ = {DI ×J (gi )} is generally reducible
I
According to Theorem 7, we have ˜ N P IJ P hk ∗ IJ χK (Ck ) χI ×J (Ck ) ΓI × ΓJ = aK ΓK where aK = | {z } K k=1 h
=χI (Ck ) χJ (Ck )
I
The multiplication table for the irreducible ΓI of G P Irepresentations J lists a Γ K K k
I
Example: Permutation group P3 χ(C) e Γ1 1 Γ2 1 Γ3 2
a, b 1 1 −1
c, d, f 1 −1 0
ΓI × ΓJ Γ1 Γ2 . . . Γ1 Γ2 .. .
ΓI × ΓJ Γ1 Γ2 Γ3 Γ1 Γ1 Γ2 Γ3 Γ2 Γ1 Γ3 Γ3 Γ1 + Γ2 + Γ3 Roland Winkler, NIU, Argonne, and NCTU 2011−2015
(Anti-) Symmetrized Product Representations Let {|σµ i} and {|τν i} be two sets of basis functions for the same n-dim. need not be representation Γ = {D(g )} with characters {χ(g )}. (again: irreducible) (1) “Simple” Product: I
|ψµν i = |σµ i|τν i,
I
ˆ )|ψµν i = P(g ≡
n n P P
(discussed previously) o total: n2
µ = 1, . . . , n ν = 1, . . . n
D(g )µ0 µ D(g )ν 0 ν |σµ0 i|τν 0 i
µ0 =1 ν 0 =1 n n P P
[D(g ) ⊗ D(g )]µ0 ν 0 ,µν |ψµ0 ν 0 i
µ0 =1 ν 0 =1 I
Character tr[D(g ) ⊗ D(g )] = χ2 (g )
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
(Anti-) Symmetrized Product Representations (cont’d) (2) Symmetrized Product: I I
µ = 1, . . . , n s total: 12 n(n + 1) |ψµν i = 12 (|σµ i|τν i + |σν i|τµ i), ν = 1, . . . µ n n P P s ˆ )|ψµν Dµ0 µ Dν 0 ν (|σµ i|τν i + |σν i|τµ i) P(g i = 21 o
µ0 =1 ν 0 =1
= ≡
n P
µ0 −1 P s s (Dµ0 µ Dν 0 ν + Dµ0 ν Dν 0 µ )|ψµ0 ν 0 i + Dµ0 µ Dµ0 ν |ψµ0 µ0 i
µ0 =1 ν 0 =1 µ0 n P P µ0 =1
I
(s)
[D(g ) ⊗ D(g )]µ0 ν 0 ,µν |ψµs 0 ν 0 i
ν 0 =1
tr[D(g ) ⊗ D(g )](s) = = = =
µ−1 n P P (Dµµ Dνν + Dµν Dνµ ) + Dµµ Dµµ µ=1 ν=1 n P n P 1 [Dµµ (g )Dνν (g ) + Dµν (g )Dνµ (g )] 2 µ=1 ν=1 n n P P 1 2 Dµµ (g ) Dνν (g ) + Dµµ (g ) 2 µ=1 ν=1 1 2 2 [χ(g )
+ χ(g 2 )] Roland Winkler, NIU, Argonne, and NCTU 2011−2015
(Anti-) Symmetrized Product Representations (cont’d) (3) Antisymmetrized Product: I I
µ = 1, . . . , n a |ψµν i = 12 (|σµ i|τν i − |σν i|τµ i), total: 12 n(n − 1) ν = 1, . . . µ − 1 n n P P a ˆ )|ψµν P(g i = 21 Dµ0 µ Dν 0 ν (|σµ i|τν i − |σν i|τµ i) o
=
µ0 =1 ν 0 =1 0 −1 n P µP
µ0 =1 ν 0 =1
(Dµ0 µ Dν 0 ν − Dµ0 ν Dν 0 µ )|ψµa 0 ν 0 i
0
≡
−1 n µP P (a) [D(g ) ⊗ D(g )]µ0 ν 0 ,µν |ψµa 0 ν 0 i
µ0 =1 ν 0 =1 I
tr[D(g ) ⊗ D(g )](a) = = = =
n µ−1 P P (Dµµ Dνν − Dµν Dνµ ) µ=1 ν=1 n P n P 1 [Dµµ (g )Dνν (g ) − Dµν (g )Dνµ (g )] 2 µ=1 ν=1 n n P P 1 2 Dµµ (g ) Dνν (g ) − Dµµ (g ) 2 µ=1 ν=1 1 2 2 [χ(g )
− χ(g 2 )] Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Intermezzo: Material Tensors to be added . . .
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Discussion I
Representation – Vector Space The matrices {D(gi )} of an n-dimensional (reducible or irreducible) representation describe a linear mapping of a vector space V onto itself. P D(gi ) u = (u1 , . . . , un ) ∈ V : u −−−→ u0 ∈ V with uµ0 = D(gi )µν uν
I
Irreducible Representation (IR) – Invariant Subspace The decomposition of a reducible representation into IRs ΓI corresponds to a decomposition of the vector space V into invariant subspaces SI such that
ν
DI (gi )
SI −−−−→ SI
∀gi ∈ G
(i.e., no mixing)
This decomposition of V lets us break down a big physical problem into smaller, more tractable problems I
Product Representation – Product Space A product representation ΓI × ΓJ describes a linear mapping of the product space SI × SJ onto itself DI ×J (gi )
SI × SJ −−−−−→ SI × SJ
I
∀gi ∈ G P IJ The block diagonalization ΓI × ΓJ = K aK ΓK corresponds to a decomposition of SI × SJ into invariant subspaces SK Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Discussion (cont’d)
Clebsch-Gordan Coefficients (CGC) I
P IJ The block diagonalization ΓI × ΓJ = K aK ΓK corresponds to a decomposition of SI × SJ into invariant subspaces SK
⇒ Change of Basis: unitary transformation IJ aK PP ( SI × SJ −→ SK` K `=1
old basis ` eK κ
Thus where
{eIµ eJν }
=
P µν
I J K` µ ν κ
` −→ new basis {eK κ }
I J K` µ ν κ
eIµ eJν
index ` not needed IJ ≤ 1 if often aK
= Clebsch-Gordan coefficients (CGC)
Clebsch-Gordan coefficients describe the unitary transformation for the decomposition of the product space SI × SJ into invariant subspaces SK` Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Clebsch-Gordan Coefficients (cont’d) Remarks I
CGC are independent of the group elements gi
I
CGC are tabulated for all important groups (e.g., Koster, Edmonds)
I
Note: Tabulated CGC refer to a particular definition (phase convention) for the basis vectors {eIµ } and representation matrices {DI (gi )}
I
Clebsch-Gordan coefficients C describe a unitary basis transformation C† C = C C† = 1
I
Thus Theorem 13: Orthogonality and completeness of CGC X I J K ` ∗ I J K 0 `0 = δKK 0 δκκ0 δ``0 µ ν κ µ ν κ0 µν X I J K ` ∗ I J K ` = δµµ0 δνν 0 µ ν κ µ0 ν 0 κ K `κ Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Clebsch-Gordan Coefficients (cont’d) Clebsch-Gordan coefficients block-diagonalize the representation matrices (unitary transformation)
111 000 111 000
11111 00000 11111 00000 (1)
11111 00000 00000 11111 00000 11111 IxJ 00000 11111 00000 11111 00000 11111 00000 11111
111 000 000 111 (2)
111 000 00 11 000 111 00 11 00 11
11 00 00 11 00 11
=C = C†
111 000 00 11 000 111 00 11 00 11
11 00 00 11 00 11
C†
11111 00000 00000 11111 11111 00000 00000 11111 00000 11111 IxJ 00000 11111 00000 11111 00000 11111 00000 11111
C
More explicitly: Theorem 14: Reduction of Product Representation ΓI × ΓJ ∗ XX I J K ` I J K ` 0 (1) DI (gi )µµ0 DJ (gi )νν 0 = µ ν κ DK (gi )κκ µ0 ν 0 κ0 K ` κκ0
(2)
DK (gi )κκ0 δKK 0δ``0 X X I J K ` ∗ I J K 0 `0 0 0 = D (g ) D (g ) 0 0 0 I i µµ J i νν µ ν κ µ ν κ µµ0 νν 0
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Evaluating Clebsch-Gordan Coefficients I
A group G is called simply reducible if its product representations IJ ΓI × ΓJ contain the IRs ΓK only with multiplicities aK = 0 or 1.
I
For simply reducible groups (⇒ no index `) according to Theorem 14 (1): nK X DI (gi )µµ0 DJ (gi )νν 0 DK∗ (gi )κ˜ κ˜ 0 h i X X I J K 0 I J K 0 ∗ nK X = DK 0 (gi )κκ0 DK∗ (gi )κ˜ κ˜ 0 µ ν κ µ0 ν 0 κ0 h 0 0 i ∗ K κκ {z } | I J I J K K 0 0 0 = δ δ δ (Theorem 4) = µ ν κ K K κ ˜κ κ ˜ κ ˜ µ0 ν 0 κ ˜0
I
Choose triple µ = µ0 = µ0 , ν = ν 0 = ν0 , and κ ˜=κ ˜ 0 = κ0 such that LHS 6= 0 r nK X ⇒ µI νJ κK = DI (gi )µ0 µ0 DJ (gi )ν0 ν0 DK∗ (gi )κ0 κ0 > 0 0 0 0 h i
Given the representation matrices {DI (g )}, the CGCs are unique for each triple I , J, K up to an overall phase that we choose such that µI 0 νJ0 κK0 > 0
I
nK X DI (gi )µµ0 DJ (gi )νν0 DK∗ (gi )κκ0 I J K h i µ0 ν0 κ0 ∀µ, ν, κ IJ If aK > 1: CGCs not unique ⇒ trickier! ⇒
I J K µ ν κ
=
1
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: CGC for group P3 ' C3v IJ ≤ 1, so we may drop the index `. This group is simply reducible, aK
Here: For Γ3 use the representation matrices {D3 (g )} corresponding to the basis functions x, y .
1 1 1 1 2 2 2 2 1 = = =1 11 1 11 1 1 1 1 1 3 3 2 3 3 1 0 0 1 = = 0 1 µν −1 0 µν 1 µ ν 1 µ ν
√ 3 3 1 0√ 1/ 2 = 0 1/ 2 µν µ ν 1 √ 3 3 3 1/ 2 0√ = 0 −1/ 2 µν µ ν 1
√ 3 3 2 0√ 1/ 2 = −1/ 2 0 µ ν 1 µν √ 3 3 3 0√ −1/ 2 = −1/ 2 0 µ ν 2 µν
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Comparison: Rotation Group I
Angular momentum j = 0, 1/2, 1, 3/2, . . . corresponds to the irreducible representations of the rotation group
I
For each j, these IRs are (2j + 1)-dimensional, i.e., the z component of angular momentum labels the basis states for the IR Γj .
I
Γj=0 is the identity representation of the rotation group
I
The product representation Γj1 × Γj2 corresponds to the addition of angular momenta j1 and j2 ; Γj1 × Γj2 = Γ|j1 −j2 | + . . . + Γj1 +j2 Here all multiplicities ajj31 j2 are one.
I
In our lecture, Clebsch-Gordan coefficients have the same meaning as in the context of the rotation group: They describe the unitary transformation from the reducible product space to irreducible invariant subspaces. This unitary transformation depends only on (the representation matrices of) the IRs of the symmetry group of the problem so that the CGC can be tabulated. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Symmetry of Observables I
ˆ Consider Hermitian operator (observable) O. Let G = {gi } be a group of symmetry transformations ˆ i )} the group of unitary operators isomorphic to G. with {P(g ˆ • For arbitrary |φi we have |ψi = O|φi. ˆ i )|ψi and |φ0 i = P(g ˆ i )|φi. • Application of gi gives |ψ 0 i = P(g ˆ i) O ˆ i )−1 ˆ 0 |φ0 i requires O ˆ 0 = P(g ˆ P(g • Thus |ψ 0 i = O
ˆ i ), O] ˆ i) O ˆ i )−1 = O ˆ =0 ˆ 0 = P(g ˆ P(g ˆ ⇔ [P(g ∀gi ∈ G If O ˆ which leaves O ˆ invariant. we call G the symmetry group of O Of course, we want the largest G possible. I
ˆ i.e., O ˆ |ni = λn |ni, Lemma: If |ni is an eigenstate of O, ˆ i ), O] ˆ i ) |ni is likewise an eigenstate of O ˆ = 0, then P(g ˆ and [P(g for the same eigenvalue λn . ˆ i ) |ni need not be orthogonal to |ni. As always P(g ˆ i ) |ni] = P(g ˆ i) O ˆ i ) |ni] ˆ [P(g ˆ |ni = λn [P(g Proof: O Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Symmetry of Observables (cont’d) I
Theorem 15: ˆ i )} be the symmetry group of the observable O. ˆ Let G = {P(g ˆ Then the eigenstates of a d-fold degenerate eigenvalue λn of O form a d-dimensional invariant subspace Sn . The proof follows immediately from the preceding lemma.
I
Most often: Sn is irreducible • central property of nature for applying group theory to physics problems • unless noted otherwise, always assumed in the following • Identify d-fold degeneracy of λn with d-dimensional IR of G.
I
Under which cirumstances can Sn be reducible? • G does not include all symmetries realized in the system, i.e., G $ G 0
(“hidden symmetry”). Then Sn is an irreducible invariant subspace of G 0 . Examples: hydrogen atom, m-dimensional harmonic oscillator (m > 1).
• A variant of the preceding case: The extra degeneracy is caused by the
antiunitary time reversal symmetry (more later). • The degeneracy cannot be explained by symmetry: rare! (Usually such “accidental degeneracies” correspond to singular points in the parameter space of a system.) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Symmetry of Observables (cont’d) Remarks: I
IRs of G give the degeneracies that may occur in the spectrum ˆ of observable O.
I
ˆ Usually, all IRs of G are realized in the spectrum of observable O ˆ form complete set) (reasonable if eigenfunctions of O
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Application 1: Symmetry-Adapted Basis ˆ = Hamiltonian ˆ=H Let O ˆ according to the IRs ΓI I Classify the eigenvalues and eigenstates of H ˆ of the symmetry group G of H. ˆ |I µ, αi = EI α |I µ, αi µ = 1, . . . , nI Notation: H
α: distinguish different levels transforming according to same ΓI
If ΓI is nI -dimensional, then eigenvalues EI α are nI -fold degenerate. Note: In general, the “quantum number” I cannot be associated directly with an observable. I
For given EI α , it suffices to calculate one eigenstate |I µ0 , αi. Then ˆ i ) |I µ0 , αi : gi ∈ G} {|I µ, αi : µ = 1, . . . , nI } = {P(g (i.e., both sets span the same subspace of H)
I
n o J = 1, . . . , N; Expand eigenstates |I µ, αi |Jν, βi : ν = 1, . . . , nJ ; in a symmetry-adapted basis β = 1, 2, . . . P P |I µ, αi = hJν, β|I µ, αi |Jν, βi = hI α||I βi |I µ, βi {z } Jν,β | β =δIJ δµν hI α||I βi
see Theorem 11
ˆ independent of specific details ⇒ partial diagonalization of H Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Application 2: Effect of Perturbations I
I
I
ˆ 0 = unperturbed Hamiltonian: H ˆ 0 |ni = En(0) |ni H ˆ 1 = perturbation H ˆ 1 |n0 i|2 P |hn|H ˆ 1 |ni + Perturbation expansion En = En(0) + hn|H + ... (0) (0) n0 6=n En − En0 ˆ 0 i = En(0) δnn0 + hn|H ˆ 1 |n0 i ⇒ need matrix elements hn|H|n ˆ0 o Let G0 = symmetry group of H usually G $ G0 ˆ G = symmetry group of H ˆ =H ˆ0 + H ˆ1, Let H
I
The unperturbed eigenkets {|ni} transform according to IRs Γ0I of G0
I
{Γ0I } are also representations of G, yet then reducible
I
Every IR Γ0I of G0 breaks down into (usually multiple) IRs {ΓJ } of G P Γ0I = aJ ΓJ (see Theorem 7) J
I
⇒ compatibility relations for irreducible representations ˆ 0 = J 0 µ0 α0 i = δJJ 0 δµµ0 hJα||H||J ˆ 0 α0 i Theorem 16: hn = Jµα|H|n ˆ i )† H ˆ P(g ˆ i) ˆ = P(g ˆ j )† H ˆ P(g ˆ j ) = 1 P P(g Proof: Similar to Theorem 11 with H i h Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Compatibility Relations for C3v ' P3 I
Character table C3v ' P3
I
C3v ' P3 has two subgroups C3 = {E , C3 , C32 = C3−1 } ' G1 = {e, a, b} Cs = {E , σv } ' G2 = {e, c} = {e, d} = {e, f }
I
Both subgroups are Abelian, so they have only 1-dim. IRs C3 E C3 C32
E E C3 C32
C3 C3 C32 E
C32 C32 E C3
C3 Γ1 Γ2 Γ3
C3v P3 Γ1 Γ2 Γ3
E C3 C32 1 1 1 1 ω ω∗ 1 ω∗ ω ω≡e 2πi/3
I
compatibility relations C3v P3 Γ1 Γ2 Γ3 C3 G1 Γ1 Γ1 Γ2 + Γ3 Cs G2 Γ1 Γ2 Γ1 + Γ2
E 2C3 3σv e a, b c, d, f 1 1 1 1 1 −1 2 −1 0
Cs E σi E E σi σi σi E C3
Cs E σi Γ1 1 1 Γ2 1 −1 C 3v
Γ2 Γ3
Γ2 2−fold degen.
Γ1
Γ2 nondegen.
Γ1
Γ2 nondegen.
Cs Γ1 Γ2 Γ2 Γ1
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Discussion: Compatibility Relations Compatibility relations and Theorem 16 tell us how a degenerate level transforming according to the IR Γ0I of G0 splits into multiple levels ˆ1 transforming according to certain IRs {ΓJ } of G when the perturbation H reduces the symmetry from G0 to G $ G0 .
Thus qualitative statements: I
ˆ1? Which degenerate levels split because of H
I
ˆ1? Which degeneracies remain unaffected by H
I
These statements do not require any perturbation theory in the conventional sense. (For every pair G0 and G, they can be tabulated once and forever!)
I
ˆ1. These statements do not require some kind of “smallness” of H
I
But no statement whether (or how much) a level will be raised ˆ1. or lowered by H Roland Winkler, NIU, Argonne, and NCTU 2011−2015
(Ir)Reducible Operators I
ˆ requires Up to now: symmetry group of operator O ˆ i) O ˆ i )−1 = O ˆ P(g ˆ P(g ∀gi ∈ G
I
ˆ ν : ν = 1, . . . , n} with More general: A set of operators {Q n ˆ i) Q ˆ i )−1 = P D(gi )µν Q ˆ ν P(g ˆµ P(g µ=1
∀ ν = 1, . . . , n ∀ gi ∈ G
is called reducible (irreducible), if Γ = {D(gi ) : gi ∈ G} is a reducible (irreducible) representation of G. ˆ i) Q ˆ i )−1 ˆ ν ≡ P(g ˆ ν P(g Often a shorthand notation is used: gi Q I
ˆ ν } transform according to Γ. We say: The operators {Q
I
ˆ ν } will not transform Note: In general, the eigenstates of {Q according to Γ. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
(Ir)Reducible Operators (cont’d) Examples: I
Γ1 = “identity representation”; D(gi ) = 1 ∀gi ∈ G; ˆ i) Q ˆ i) ˆ P(g ⇒ P(g
−1
ˆ =Q
nI = 1
∀gi ∈ G
ˆ is a scalar operator or invariant. We say: Q I
ˆ most important scalar operator: the Hamiltonian H ˆ i.e., H always transforms according to Γ1 ˆ is the largest symmetry group The symmetry group of H ˆ that leaves H invariant.
I
position operator xˆν momentum operator pˆν = −i~ ∂xν ⇒
I
ν = 1, 2, 3
(polar vectors)
{ˆ xν } and {ˆ pν } transform according to 3-dim. representation Γpol (possibly reducible!)
composite operators (= tensor operators) P e.g., angular momentum ˆlν = ελµν xˆλ pˆµ λ,µ
ν = 1, 2, 3
(axial vector)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Tensor Operators I
I
ˆ I ≡ {Q ˆ Iµ : µ = 1, . . . , nI } transform according to ΓI = {DI (gi )} Let Q J ˆ ˆ Jν : ν = 1, . . . , nJ } transform according to ΓJ = {DJ (gi )} Q ≡ {Q I J µ = 1, . . . , n I ˆµ Q ˆν : transforms according to the product Then Q ν = 1, . . . , nJ representation ΓI × ΓJ
I
ΓI × ΓJ is, in general, reducible I J ˆ ν is likewise reducible ˆµ Q ⇒ The set of tensor operators Q
I
A unitary transformation brings ΓI × ΓJ = {DI (gi ) ⊗ DJ (gi )} into block-diagonal form I J ˆµ Q ˆ ν into irreducible tensor ⇒ The same transformation decomposes Q operators (use CGC)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Where Are We? We have discussed I
the transformational properties of states
I
the transformational properties of operators
Now: I
the transformational properties of matrix elements
⇒ Wigner-Eckart Theorem
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Wigner-Eckart Theorem Let {|I µ, αi : µ = 1, . . . , nI } transform according to ΓI = {DI (gi )} {|I 0 µ0 , α0 i : µ0 = 1, . . . , nI 0 } transform according to ΓI 0 = {DI 0 (gi )} ˆ J = {Q ˆ Jν : ν = 1, . . . , nJ } transform according to ΓJ = {DJ (gi )} Q XJ I I0 ` 0 0 ˆ Jν | I µ, αi = ˆJ Then hI 0 µ0 , α0 | Q ν µ µ0 hI α || Q || I αi` `
ˆ J || I αi is independent where the reduced matrix element hI 0 α0 || Q ` 0 of µ, µ and ν. Proof: I
I
ˆ Jν |I µ, αi : µ = 1, . . . , nI Q ν = 1, . . . , nJ
transforms according to ΓI × ΓJ P J I K` J ˆ Thus CGC expansion Qν |I µ, αi = |K κ, `i ν µ κ K κ,`
I
ˆ Jν | I µ, αi = hI 0 µ0 , α0 | Q
P K κ,`
J I K` ν µ κ
hI 0 µ0 , α0 | K κ, `i Theorem 11 | {z } ˆ J || I αi ≡ δI 0 K δµ0 κ hI 0 α0 || Q ` Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Discussion: Wigner-Eckart Theorem I
Matrix elements factorize into two terms • the reduced matrix element independent of µ, µ0 and ν • CGC indexing the elements µ, µ0 and ν of ΓI , ΓI 0 and ΓJ .
ˆ J) (CGC are tabulated, independent of Q I
Thus:
I
Matrix elements for different values of µ, µ0 and ν have a fixed ratio ˆJ independent of Q
I
If ΓI 0 is not contained in ΓI × ΓJ ⇒
reduced matrix element = “physics” Clebsch-Gordan coefficients = “geometry”
J I I0 ` ν µ µ0
=0
Equivalent to: If Γ∗I 0 × ΓJ × ΓI does not contain the identity representation
∀ ν, µ, µ0
ˆ Jν | I µ, αi = 0 ⇒ hI 0 µ0 , α0 | Q
∀ ν, µ, µ0
Many important selection rules are some variation of this result. I
Theorems 11 and 16 are special cases of the WE theorem for ˆ ˆ 1 = 1 and Q ˆ1 = H Q (yet we proved the WE theorem via Theorem 11) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Discussion: Wigner-Eckart Theorem (cont’d) Application: Perturbation theory I
Compatibility relations and Theorem 16 describe splitting of degenerate levels using the symmetry group G of perturbed problem
I
Alternative approach Splitting of levels using the symmetry group G0 of unperturbed problem (i.e., no need to know group G of perturbed problem) ˆ J be tensor operator transforming according to IR ΓJ of G0 • Let Q ˆ1 = F J · Q ˆ 1 is proportional to ˆ J = FνJ Q ˆ Jν i.e., H • Often: perturbation H ˆJ only νth component of tensor operator Q ˆ J projected on component Q ˆ Jν via suitable orientation of field F J Q
ˆ =H ˆ0 + H ˆ 1 is subgroup G ⊂ G0 which leaves Q ˆ Jν invariant. • Symmetry group of H • WE Theorem:
ˆ 1 |n0 i = FνJ hn=I µα|Q ˆ Jν |n0 =I 0 µ0 α0 i = hn|H
P `
J I0 I ` ν µ0 µ
ˆ J || I 0 α0 i hI α || Q `
• Changing the orientation of F J changes only the CGCs in (∗)
ˆ J || I 0 α0 i are “universal” The reduced matrix elements hI α || Q ` Roland Winkler, NIU, Argonne, and NCTU 2011−2015
(∗)
Example: Optical Selection Rules Example: Optical transitions for a system with symmetry group C3v (e.g., NH3 molecule) I
Optical matrix elements hiI |e · ˆr|fJ i (dipole approximation) where
|iI i = initial state (with IR ΓI ); |fJ i = final state (IR ΓJ ) e = (ex , ey , ez ) = polarization vector ˆr = (ˆ x , yˆ , zˆ) = dipole operator (≡ position operator)
I
xˆ, yˆ transform according to Γ3 zˆ transforms according to Γ1
I
e.g., light xy polarized: hi1 |ex xˆ + ey yˆ |f3 i • transition allowed because Γ3 × Γ3 = Γ1 + Γ2 + Γ3 • in total 4 different matrix elements, but only one reduced matrix
element I
z polarized: hi1 |ez zˆ|f3 i • transition forbidden because Γ1 × Γ3 = Γ3
I
These results are independent of any microscopic models for the NH3 molecule! Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Goal: Spin 1/2 Systems and Double Groups
Rotations and Euler Angles I I
I I
So far: transformation of functions and operators dependent on position Now: systems with spin degree of freedom ⇒ wave functions are two-component Pauli spinors ψ↑ (r) Ψ(r) = ψ↑ (r) |↑i + ψ↓ (r) |↓i ≡ ψ↓ (r) How do Pauli spinors transform under symmetry operations? Parameterize rotations via Euler angles α, β, γ z
z
z z’
y’
z y’
β
y
z’
y’
z’ z
z y’’ y’
β
y
y x’’’ γ x’’
x’’ x’
x’ α
x
axis z, angle α I
y’
α
x
axis y’, angle β
x’ α
x
axis z’, angle γ
Thus general rotation R(α, β, γ) = Rz 0 (γ) Ry 0 (β) Rz (α) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Rotations and Euler Angles (cont’d) I
General rotation R(α, β, γ) = Rz 0 (γ) Ry 0 (β) Rz (α)
I
Difficulty: axes y 0 and z 0 refer to rotated body axes (not fixed in space)
I
Use Rz 0 (γ) = Ry 0 (β) Rz (γ) Ry−1 0 (β) −1 Ry 0 (β) = Rz (α) Ry (β) Rz (α)
I
Thus R(α, β, γ) = Ry 0 (β) Rz (γ) Ry−1 0 (β) Ry 0 (β) Rz (α) | {z } | {z }
preceding rotations are temporarily undone
=1
Rz (α) Ry (β) Rz−1 (α)
rotations about z axis commute
rotations about space-fixed axes!
I
Thus R(α, β, γ) = Rz (α) Ry (β) Rz (γ)
I
More explicitly: rotations of (x, y , z) ∈ R3 vectors r = cos α − sin α 0 Rz (α) = sin α cos α 0 , 0 0 1
I
I
cos β 0 − sin β 1 0 Ry (β) = 0 sin β 0 cos β
etc.
SO(3) = set of all rotation matrices R(α, β, γ) = set of all orthogonal 3 × 3 matrices R with det R = +1. R(2π, 0, 0) = R(0, 2π, 0) = R(0, 0, 2π) = 1 ≡ e Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Rotations: Spin 1/2 Systems I
I
Rotation matrices for spin-1/2 spinors (axis n) Rn (φ) = exp − 2i σ · n φ = 1 cos(φ/2) − in · σ sin(φ/2) R(α, β, γ) = Rz (α) Ry (β) Rz (γ) −i(α+γ)/2 e cos(β/2) −e −i(α−γ)/2 sin(β/2) = e −i(α−γ)/2 sin(β/2) e i(α+γ)/2 cos(β/2) transformation matrix for spin 1/2 states
I
SU(2) = set of all matrices R(α, β, γ) = set of all unitary 2 × 2 matrices R with det R = +1.
I
R(2π, 0, 0) = R(0, 2π, 0) = R(0, 0, 2π) = −1 ≡ e¯
rotation by 2π is not identity
I
R(4π, 0, 0) = R(0, 4π, 0) = R(0, 0, 4π) = 1 = e
rotation by 4π is identity
I
Every SO(3) matrix R(α, β, γ) corresponds to two SU(2) matrices R(α, β, γ) and R(α + 2π, β, γ) = R(α, β + 2π, γ) = R(α, β, γ + 2π) = e¯ R(α, β, γ) = R(α, β, γ) e¯ ⇒ SU(2) is called double group for SO(3) Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Double Groups I
Definition: Double Group Let the group of spatial symmetry transformations of a system be G = {gi = R(αi , βi , γi ) : i = 1, . . . , h} ⊂ SO(3) Then the corresponding double group is Gd =
{gi = R(αi , βi , γi ) : i = 1, . . . , h} ∪ {gi = R(αi + 2π, βi , γi ) : i = 1, . . . , h} ⊂ SU(2)
I
Thus with every element gi ∈ G we associate two elements gi and g¯i ≡ e¯ gi = gi e¯ ∈ Gd
I
If the order of G is h, then the order of Gd is 2h.
I
Note: G is not a subgroup of Gd because the elements of G are not a closed subset of Gd . Example: Let g = rotation by π • in G: g 2 = e • in Gd : g 2 = e ¯
I
the same group element g is thus interpreted differently in G and Gd
Yet: {e, e¯} is invariant subgroup of Gd and the factor group Gd /{e, e¯} is isomorphic to G. ⇒ The IRs of G are also IRs of Gd
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Double Group C3v C3v Γ1 Γ2 Γ3 Γ4 Γ5 Γ6
¯3 3σv 3¯ E E¯ 2C3 2C σv 1 1 1 1 1 1 1 1 1 1 −1 −1 2 2 −1 −1 0 0 2 −2 1 −1 0 0 1 −1 −1 1 i −i 1 −1 −1 1 −i i
I
For Γ1 , Γ2 , and Γ3 the “barred” group elements have the same characters as the “unbarred” elements. Here the double group gives us the same IRs as the “single group”
I
For other groups a class may contain both “barred” and “unbarred” group elements. ⇒ the number of classes and IRs in the double group need not be twice the number of classes and IRs of the “single group” Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Time Reversal (Reversal of Motion) I I
Time reversal operator θˆ : t → −t ˆ Action of θ: θˆˆr θˆ−1 = ˆr independent of t −1 ˆ θˆ = −ˆ θˆ p p ˆ θˆ−1 = −L ˆ θˆ L −1 ˆ ˆ ˆ ˆ θ S θ = −S
I
linear in t
ˆ δt/~ ˆ Consider time evolution: U(δt) = 1 − iH ˆ ˆ ⇒ U(δt) θˆ |ψi = θˆ U(−δt) |ψi ⇔ ⇒
ˆ θˆ |ψi = θˆ i H ˆ |ψi −i H Need
θˆ = UK
but need also
ˆ H] ˆ =0 [θ,
U = unitary operator K = complex conjugation
Properties of θˆ = UK : I K c1 |αi + c2 |βi = c ∗ |αi + c ∗ |βi (antilinear) 1 2 ˜ = θˆ |βi I Let |˜ αi = θˆ |αi and |βi ˜ αi = hβ|αi∗ = hα|βi ⇒ hβ|˜
θˆ = UK is antiunitary operator
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Time Reversal (cont’d) The explicit form of θˆ depends on the representation I position representation: θˆˆr θˆ−1 = ˆr ⇒ θˆ ψ(r) = ψ ∗ (r) ˆp I momentum representation: θ ˆ θˆ−1 = −ˆ p ⇒ θˆ ψ(p) = ψ ∗ (−p) I
spin 1/2 systems: • θˆ = iσy K ⇒ θˆ2 = −1 ˆ are at least two-fold degenerate • all eigenstates |ni of H (Kramers degeneracy)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Time Reversal and Group Theory I
ˆ Consider a system with Hamiltonian H.
I
ˆ Let G = {gi } be the symmetry group of spatial symmetries of H ˆ i ), H] ˆ = 0 ∀gi ∈ G [P(g
I
ˆ Let {|I νi : ν = 1, . . . , nI } be an nI -fold degenerate eigenspace of H which transforms according to IR ΓI = {DI (gi )} ˆ |I νi = EI |I νi ∀ν H P ˆ i ) |I νi = P(g DI (gi )µν |I µi
I
ˆ be time-reversal invariant: [H, ˆ θ] ˆ =0 Let H ˆ ˆ i )}) with ⇒ θ is additional symmetry operator (beyond {P(g ˆ P(g ˆ i )] = 0 [θ,
I
ˆ i ) θˆ |I νi = θˆ P(g ˆ i )|I νi = θˆ P DI (gi )µν |I µi = P DI∗ (gi )µν θˆ |I µi P(g
I
Thus: time-reversed states {θˆ |I νi} transform according to complex conjugate IR Γ∗I = {DI∗ (gi )}
µ
µ
µ
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Time Reversal and Group Theory (cont’d) I
time-reversed states {θˆ |I νi} transform according to complex conjugate IR Γ∗I = {DI∗ (gi )}
Three possiblities (known as “cases a, b, and c”) (a) {|I νi} and {θˆ |I νi} are linear dependent (b) {|I νi} and {θˆ |I νi} are linear independent The IRs ΓI and Γ∗I are distinct, i.e., χI (gi ) 6= χ∗I (gi ) (c) {|I νi} and {θˆ |I νi} are linear independent ΓI = Γ∗I , i.e., χI (gi ) = χ∗I (gi ) ∀ gi Discussion I Case (a): time reversal is additional constraint for {|I νi} e.g., nI = 1 ⇒ |νi reell I
Cases (b) and (c): time reversal results in additional degeneracies
I
Our definition of cases (a)–(c) follows Bir & Pikus. Often (e.g., Koster) a different classification is used which agrees with Bir & Pikus for spinless systems. But cases (a) and (c) are reversed for spin-1/2 systems. Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Time Reversal and Group Theory (cont’d) I
When do we have case (a), (b), or (c)?
Criterion by Frobenius & Schur η case (a) 1P 2 0 case (b) χI (gi ) = h i −η case (c) where η =
+1 systems with integer spin −1 systems with half-integer spin
Proof: Tricky! (See, e.g., Bir & Pikus)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Example: Cyclic Group C3 I
C3 is Abelian group with 3 elements: C3 = {q, q 2 , q 3 ≡ e}
I
Multiplication table 2
C3 e q q e e q q2 q q q2 e q2 q2 e q
Character table C3 e
q
q2
Γ1 1 1 1 Γ2 1 ω ω ∗ Γ3 1 ω ∗ ω
time reversal
a b b
ω≡e 2πi/3 I
IR Γ1 : no additional degeneracies because of time reversal
I
IRs Γ2 and Γ3 : these complex IRs need to be combined ⇒ two-fold degeneracy because of time reversal symmetry (though here no spin!)
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Group Theory in Solid State Physics First: Some terminology I
Lattice: periodic array of atoms (or groups of atoms)
I
Bravais lattice: Rn = nx ax + ny ay + nz az
n = (n1 , n2 , n3 ) ∈ Z3 ai linearly independent
Every lattice site Rn is occupied with one atom Example: 2D honeycomb lattice is not a Bravais lattice I
Lattice with basis: • Every lattice site Rn is occupied with z atoms • Position of atoms relative to Rn :
τi,
i = 1, . . . , q
• These q atoms with relative positions τ i form a basis. • Example: two neighboring atoms in 2D honeycomb lattice
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Symmetry Operations of Lattice I
Translation t
(not necessarily by lattice vectors Rn ) →
3 × 3 matrices α
I
Rotation, inversion
I
Combinations of translation, rotation, and inversion ⇒ general transformation for position vector r ∈ R3 : r0 = αr + t ≡ {α|t} r
I
Notation {α|t} includes also • Mirror reflection = rotation by π about axis perpendicular
to mirror plane followed by inversion • Glide reflection = translation followed by reflection • Screw axis = translation followed by rotation
Symmetry operations {α|t} form a group I
Multiplication {α0 |t0 } {α|t} r = α0 r0 + t0 = α0 αr + α0 t + t0 | {z } = {α0 α|α0 t + t0 } r r0 =αr+t
I
Inverse Element {α|t}−1 = {−α−1 | − α−1 t}
because {α|t}−1 {α|t} = {α−1 α|α−1 t − α−1 t} = {1|0} Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Classification
Symmetry Groups of Crystals to be added . . .
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Symmetry Groups of Crystals
Translation Group Translation group = set of operations {1|Rn } I
{1|Rn0 } {1|Rn } = {1|Rn0 + 1 Rn } = {1|Rn0 +n } ⇒ Abelian group
I
associativity (trivial)
I
identity element {1|0} = {1|R0 }
I
inverse element {1|Rn }−1 = {1| − Rn }
Translation group Abelian ⇒ only one-dimensional IRs
Roland Winkler, NIU, Argonne, and NCTU 2011−2015
Irreducible Representations of Translation Group (for clarity in one spatial dimension) I I
I
I
Consider translations {1|a} ˆa = T ˆ{1|a} is unitary operator Translation operator T ⇒ eigenvalues have modulus 1 ˆa |φi = e −iφ |φi −π < φ ≤ π eigenvalue equation T ˆna |φi = e −inφ |φi more generally T n∈Z ⇒ representations D({1|Rna }) = e −inφ
−π
