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/phys30101http://nuclear.ph.man.ac.uk/~jb/phys30101 Lecture 12

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. Last lecture we found operators for L 2 and L z in spherical polar coordinates: Eignefunctions could be found by separation of variables: The eigenfunctions are called Spherical Harmonics. The eigenvalues are:

4. TODAY: 3.2 Finding eigenfunctions and eigenvalues is a more abstract way using the ladder operators. 3.3 We show states of definite eigenvalue L z have axial symmetry. 3.4 Coefficients connected to the ladder operators 4. Spin – intrinsic spin= ½ħ angular momentum of electron, proton and neutron.

5. Using the ladder operators we will show: And find the commutation relations: So starting with the eigenvalue equation φ = eigenfunction, β =eigenvalue (=mħ) we can find other eigenfunctions with eignevalues one unit of ħ different from β i.e. (m+1)ħ and (m-1)ħ

6. Possible orientations of the l=2 angular momentum vector when the z-component has a definite value.

7. 4. Spin 4.1 Commutators, ladder operators, eigenfunctions, eigenvalues 4.2 Dirac notation (simple shorthand – useful for “spin” space) 4.3 Matrix representations in QM; Pauli spin matrices 4.4 Measurement of angular momentum components: the Stern-Gerlach apparatus