Symmetries in Physics 5A1335
Course information 2005
Lecturer: Håkan Snellman
- phone 5537 8172
Lecture notes are sold in the student office for 50 SEK.
The first 8 chapters of the following book covers essentially
the same material:
Solution of homework problems: There will be two sets of 4 - 5 problems each.
More information is given in the first set
Contents of the lectures
These notes will be continuously updated. Note the online introduction. Lectures 1 - 2 Introduction to symmetries, with illustrations. Symmetry transformations of geometrical figures and their group property. C3v as symmetry group of NH3. Homomorphisms and isomorphisms of groups. Groups acting on themselves by left and right multiplication. Abelian groups, additive notation. The real line R, unit circle S^1, and the integers Z as Abelian groups. Action of the group on functions defined on the group and unitary representations for Abelian groups. The Hilbert space of square integrable functions. Fourier series as expansions in this Hilbert space for S^1. A review of operators in Hilbert space, using the Dirac bra-ket notation. Hermitian, unitary and normal operators. The eigenvalues and spectral representations of operators. Functions of operators, the exponential. Lecture notes: 1.1 - 1.4. Appendix A.1-A.4. Lectures 3 - 4 Simultaneous diagonalization of commuting operators. Consequences for unitary representations of Abelian groups. Discrete Fourier transform and Fourier analysis as group representations. The dual group. Translations on Z and the corresponding unitary operators.Simplest examples of symmetry of quantum systems: mirror and inversion symmetry. Definition of a parity operator in the Hilbert space of a Schrödinger particle. The eigenvalues of the parity operator, its action on observables. The parity selection rule. The parity as a symmetry of the Hamiltonian. Schrödinger particle in a periodic potential and Bloch's theorem. The group of translations of the real line. Its action on the canonical coordinate and momentum observables. Direct products of groups. Subgroups and invariant subgroups. Lecture notes: Appendix A.3, 1.4 - 2.4.. Lectures 5 - 6 Orthogonal transformations of R^2 as simplest non-Abelian group. Unitary representations of SO(2) and O(2). The groups SO(3) and O(3). Parametrization of SO(3). The spherical harmonics as a basis of reps of O(3). Action of the orthogonal group on the Hilbert space of a spinless Schrödinger particle. Action on the coordinate and momentum vector operators and on the Hamiltonian. Central potentials, and spherical symmetry. Separation of variables and radial Hamiltonian. Selection rules and degeneracy of energy spectra due to spherical symmetry. Conjugation of group elements and decomposition of a group into conjugation classes. Example: the classes of SO(3). Permutation groups. Decomposition of a permutation into cycles. Young diagrams and the conjugation classes of the permutation group. The parity of a permutation. The alternating group of even parity permutations. Cayley's theorem. Example: the symmetry group of a regular tetrahedron as a permutation group on the vertices. The classes of the tetrahedral groups. Left cosets and Lagrange's theorem. Examples of Lagrange's theorem. Lecture notes: 3.1 - 3.4, 3.6. 4.1 - 4.4. Take a look at 3.5 for applications. Lectures 7 - 8 Normal subgroups and the factor (quotient) group. The homomorphism theorem. G-spaces, G-modules and representations. Direct sum and direct product of representations. Transitive action on G-spaces and orbits. The isotropy (stabilizer) group. The orbit-stabilizer theorem = Theorem 5.1(1). Examples from regular polyhedra. The counting lemma = Theorem 5.1(2). Example: action of the icosahedral group on the C60 molecule. Lecture notes: 4.5 - 4.6, 5.3. Take a look at 4.7, 5.1 - 5.2. Lectures 9 - 10 Euclidean group and semidirect products. Unitary reps defined by a G-space. Unitary representations, unitary equivalence. Invariant subspaces and reduction of reps. Irreps and their equivalence classes. The character of a rep and its properties. The orthogonality properties of primitive charaters. Expansion of class functions into series of characters. Reductiomn of a rep into irreps using the primitive characters. Application to some point groups. Lecture notes: 5.4, 6.1 - 6.6 Lectures 11 - 12 Proof of Schur's lemmas. Consequences of Schur's lemmas: symmetry and degeneracy, with some examples. Lifting of degeneracy when a symmetry is broken. Idea of the great orthogonality theorem, and the orthogonality of primitive characters. The regular representation. Example: symmetry and the vibration modes of a molecule. The Clebsch-Gordan series. Irreps of the permutation groups. Lecture notes: 6.7 - 6.9 Lectures 13 - 15 Lie groups. One-parameter Lie groups. Abelian Lie groups. Matrix groups and continuity in matrix norm. Continuous paths in parameter space, connected and simply connected groups. Contraction of closed paths and homotopy, the fundamental group. SO(3) has a fundamental group of order 2. Parameter space for SU(2), which is connected and simply connected. The homomorphism from SU(2) to SO(3) with kernel = the center of SU(2). The infinitesimal generators and their commutation relations. Lie algebras. The identity of the Lie algebras su(2) and so(3). Matrix groups and matrix Lie algebras. Lie groups generated by Lie algebras. The universal covering group. Representations of Lie algebras and the corresponding rep of the Lie group. Multiple-valued reps of the group. The adjoint rep, example su(2). Reps of Lie algebras: irreducibility, invariant subspaces, direct sum, tensor product. Analysis on SU(2): classes, primitive characters, invariant measure on the parameter space. Reps of the Lie algebra su(2) and the relation with the representations of the Lie group. The invariant measure on SU(2) and the regular representation. Primitive characters for SU(2) and their orthogonality properties. The completeness properties of the matrix elements of the irreps and of the characters. The Clebsch-Gordan series for SU(2). Definition and orthogonality properties of the Clebsch-Gordan coefficients. Transformation properties of observables: scalars, vector operators. Definition of irreducible tensor operators. The Wigner-Eckart theorem with applications. Representations of the permutation group. Angular momentum and multi-particle states. SU(3) symmetry and the quark model. Space-time symmetry groups, Euclidean, Lorentz and Poincaré groups. Induced irreps of the Euclidean group. Lecture notes: Ch 7 - 9