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 Ro ssler. from whom I learned group theory,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 c ab G closure. 2 a b c G ab c a bc associativity,3 e G ae a a G identity neutral element. 4 a G b G a b e i e b a inverse element,Corollaries. b a 1 a a a 1 e a G left inverse right inverse,c e a ae a a G left neutral right neutral. d a b G c ab c b 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. 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 2 i 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 Rn continuous translation group. vector addition in R,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. 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. Roland Winkler NIU Argonne and NCTU 2011 2015,Group Theory in Physics. Classical Mechanics,I Lagrange function L q q,I Lagrange equations i 1 N. I If for one j 0 pj is a conserved quantity,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,I overlapp i x j x dx ij i j e o. I expectation value hi x ii 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 Schro dinger equation 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 O,can be solved simultaneously. eigenvalue O of O is good quantum number for,Example H atom. 2 2 L 2 e2,group SO 3,2m r r r 2mr r,L 2 H L z H L 2 L z 0. eigenstates nlm r index l L 2 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 R lattice vectors,translation operator T R T R f r f r R. Bloch theorem k 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. ii Phonons,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. I SU 2 proton 12 1,2 i 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 P3 e a b c d f,e e a b c d f. 1 2 3 1 2 3 1 2 3 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 d d f c b e a. 1 3 2 3 2 1 2 1 3,f f c d a b e,I e e a b e c e d e f G are subgroups of G. Specialized Literature I G L Bir und G E Pikus Symmetry and Strain Induced E ects in Semiconductors Wiley New York 1974 thorough discussion of group theory

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