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 19

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. Plan: Include coupling of orbital and spin angular momenta in Hamiltonian for hydrogen atom L S

4. 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

5. Recap: Brief reminder of QM description of hydrogen atom TISE V(r) = -e 2 /4π ε o r Remember Angular part written: Solve by separation of variables: To get radial equation

6. Eigenenergies E n = E 1 /n 2 where E 1 = -13.6 eV n = 1, 2, 3, 4…. Allowed values of l : l = 0, 1, 2, 3…, (n-1). Allowed values of l z : m l = + l, l -1, l -2,…., 0, -1, …, - l (2 l +1 values)