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PHYS 30101 Quantum Mechanics Lecture 15 Dr Sean Freeman Nuclear Physics Group These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101 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 Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical harmonics Yl,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 Jz and J2. 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: Equation (1) Gives: Multiply from left and integrate: ) (We use And find: Exactly the rule for multiplying matrices! 4.3.2 Matix representations of Sx, Sy, Sz Sx = ½ħ σx ; Pauli Spin Matrices: Sy = ½ħ σy ; Sz = ½ħ σz Eigenfunctions of spin-1/2 operators Matrix representation: Eigenvectors of Sx, Sy, Sz 4.3.3 Example: description of spin=1 polarised along the x-axis is In Dirac notation: The Stern-Gerlach apparatus The Stern-Gerlach apparatus z y x Successive measurements on spin-1/2 particles 1 Unpolarised 1/2 1/4 Select mz=+1/2 Measure Sz Select mx=+1/2 Measure Sx 1/8 Select mz=+1/2 Measure Sz