PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Sean Freeman Nuclear Physics Group These slides at: Lecture 15

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.

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

Recap: 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:

Recap: 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)

Pauli Spin Matrices: Matix representations of S x, S y, S z S x = ½ħ σ x ; S y = ½ħ σ y ; S z = ½ħ σ z

Matrix representation: Eigenvectors of S x, S y, S z Eigenfunctions of spin-1/2 operators Example: description of spin=1 polarised along the x-axis In Dirac notation: is

The Stern-Gerlach apparatus

z x 1 1/2 1/41/8 Unpolarised Measure S z Select m z =+1/2 Measure S x Select m x =+1/2 Measure S z Select m z =+1/2 y Successive measurements on spin-1/2 particles