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Introduction to Group Theory

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Introduction to Group Theory 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 = fa; b; c;:::g is called a group, if there exists a group multiplication connecting the elements in G in the following way (1) a; b 2 G : c = a b 2 G (closure) (2) a; b; c 2 G :(ab)c = a(bc) (associativity) (3) 9 e 2 G : a e = a 8a 2 G (identity / neutral element) (4) 8a 2 G 9 b 2 G : a b = e; i.e., b ≡ a−1 (inverse element) Corollaries (a) e−1 = e (b) a−1a = a a−1 = e 8a 2 G (left inverse = right inverse) (c) e a = a e = a 8a 2 G (left neutral = right neutral) (d) 8a; b 2 G : c = a b , c−1 = b−1a−1 Commutative (Abelian) Group (5) 8a; b 2 G : a b = b a (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 Qnf0g with multiplication (Abelian group, infinite order) I complex numbers fexp(2πi m=n): m = 1;:::; ng 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! n I (continuous) translations in R : (continuous) translation group ≡ vector addition in Rn I 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 Hamiltonian H^ must be invariant under the symmetries of a quantum system ) construct H^ via symmetry arguments I ::: Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Classical Mechanics I Lagrange function L(q; q_ ), d @L @L I Lagrange equations = i = 1;:::; N dt @q_ i @qi @L @L I If for one j : = 0 ) pj ≡ is a conserved quantity @qj @q_ j Examples I qj linear coordinate I qj angular coordinate • translational invariance • rotational invariance • linear momentum pj = const. • angular momentum pj = const. • translation group • 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 (x) e (x) even: e (x) = e (−x) o (x) odd: o (x) = − o (−x) R ∗ I overlapp i (x) j (x) dx = δij i; j 2 fe; og R ∗ I expectation value hijxjii = i (x) x i (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 Schr¨odingerequation H = E or i~@t = H ^ ^ ^ ^ ^ I Let O with i~@t O = [O; H] = 0 ) O is conserved quantity ^ ^ ) eigenvalue equations H = E and O = λO^ can be solved simultaneously ^ ) eigenvalue λO^ of O is good quantum number for Example: H atom 2 @2 2 @ L^2 e2 H^ = ~ + + − ) group SO(3) I 2m @r 2 r @r 2mr 2 r 2 2 ) [L^ ; H^ ] = [L^z ; H^ ] = [L^ ; L^z ] = 0 2 ) eigenstates nlm(r): index l $ L^ ; m $ L^z 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) 8 R 2 flattice vectorsg ) translation operator T^R : T^R f (r) = f (r + R) [T^R; H^ ] = 0 ik·r ) Bloch theorem k(r) = e uk(r) with uk(r + R) = uk(r) ) wave vector k is quantum number for the discrete translation invariance, k 2 first Brillouin zone Roland Winkler, NIU, Argonne, and NCTU 2011−2015 Group Theory in Physics Quantum Mechanics (3) Solid State Physics (ii) Phonons rotation by 90o 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 Neutron mn = 939:57 MeV ∼ degeneracy I Symmetry: isospin ^I with [^I ; H^strong] = 0 1 1 1 1 I SU(2): proton j 2 2 i, neutron j 2 − 2 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 P3 e a b c d f Example: permutation group P3 e e a b c d f 1 2 3 1 2 3 1 2 3 e = a = b = a a b e f c d 1 2 3 2 3 1 3 1 2 b b e a d f c 1 2 3 1 2 3 1 2 3 c c d f e a b c = d = f = 1 3 2 3 2 1 2 1 3 d d f c b e a f f c d a b e I feg, fe; a; bg, fe; cg, fe; dg, fe; f g, G are subgroups of G Roland Winkler, NIU, Argonne, and NCTU 2011−2015 P3 e a b c d f Conclusions from e e a b c d f Group Multiplication Table a a b e f c d b b e a d f c I Symmetry w.r.t. main diagonal ) group is Abelian c c d f e a b d d f c b e a n I order n of g 2 G: smallest n > 0 with g = e f f c d a b e 2 n I fg; g ;:::; g = eg with g 2 G is Abelian subgroup (a cyclic group) I in every row / column every element appears exactly once because: Rearrangement Lemma: for any fixed g 0 2 G, we have G = fg 0g : g 2 Gg = fgg 0 : g 2 Gg i.e., the latter sets consist of the elements in G rearranged in order.
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