1. PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) j.billowes@manchester.ac.uk These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101 Lecture 18

2. Syllabus 1.Basics of quantum mechanics (QM) Postulate, operators, eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent Schrödinger equation, probabilistic interpretation, compatibility of observables, the uncertainty principle. 2.1-D QM Bound states, potential barriers, tunnelling phenomena. 3.Orbital angular momentum Commutation relations, eigenvalues of L z and L 2, explicit forms of L z and L 2 in spherical polar coordinates, spherical harmonics Y l,m. 4.Spin Noncommutativity of spin operators, ladder operators, Dirac notation, Pauli spin matrices, the Stern-Gerlach experiment. 5.Addition of angular momentum Total angular momentum operators, eigenvalues and eigenfunctions of J z and J 2. 6.The hydrogen atom revisited Spin-orbit coupling, fine structure, Zeeman effect. 7.Perturbation theory First-order perturbation theory for energy levels. 8.Conceptual problems The EPR paradox, Bell’s inequalities.

3. Coupling two angular momenta L S When M (= m 1 + m 2 ) is a constant of motion, m 1 and m 2 are not well defined

4. We shall try and follow this convention: Capitals J, L, S indicate angular momentum vectors with magnitudes that can be expressed in units of ħ: L 2 = l ( l + 1) ħ 2 Lower case j, l, s indicate quantum numbers that are integer or half-integer: l = 0, 1, 2, 3… s = 1/2 j = 1/2, 3/2, 5/2 Lower case vectors j, l, s indicate vectors whose components along a quantization axis are integer or half- integer values (ie not in units of ħ).

5. We’ll do this now

6. Choice of basis for atomic electrons Clebsch-Gordan coefficients

9. 6. The hydrogen atom revisited - Reminder of eigenfunctions, eigenvalues and quantum numbers n, l, m l of hydrogen atom. 6.1 Spin-orbit coupling and the fine structure. 6.2 Zeeman effect for single electron atoms in (a) a weak magnetic field (b) a strong magnetic field 6.3 Spin in magnetic field: QM and classical descriptions