1. PHYS 30101 Quantum Mechanics PHYS 30101 Quantum Mechanics Dr Gavin Smith Nuclear Physics Group These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101 Lecture 14

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. 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

4. RECAP: 4. Spin (algebra almost identical to orbital angular momentum algebra – except we can’t write down explicit analogues of spherical harmonics for spin eigenfunctions) Commutation relations (plus two others by cyclic permutation of x,y,z) By convention we choose to work with eigenfunctions of S 2 and S z which we label α and β So, the eigenvalue equations are:

5. Any general spin-1/2 wavefunction χ can be written as a linear combination of the complete set of our chosen eigenfunction set: χ = a α+ b β The coefficients a and b give the weighting and relative phases of the α and β eigenstates. Normalization: a 2 + b 2 = 1 The wavefunction χ could be, for example, that of a spin-1/2 particle polarised in the x-direction (an eigenstate of S x ) We now find the coefficients a, b for this state as an example (there’s only two eigenfunctions in the set)

6. χ = a α+ b β Eigenfunctions and eigenvalues of S x, S y, S z described in this way:

7. RECAP: 4.2 Dirac notation Dirac

8. 4.3 Matrix representations in QM We can describe any function as a linear combination of our chosen set of eigenfunctions (our “basis”) Substitute in the eigenvalue equation for a general operator: Gives:

9. 4.3 Matrix representations in QM We can describe any function as a linear combination of our chosen set of eigenfunctions (our “basis”) Substitute in the eigenvalue equation for a general operator: Gives: Multiply from left and integrate: (We use ) And find: Exactly the rule for multiplying matrices! Equation (1)

10. Matrix representation: Eigenvectors of S x, S y, S z Eigenfunctions of spin operators (from lecture 13)