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PHYS 30101 Quantum Mechanics Lecture 17 Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) [email protected] These slides at: http://nuclear.ph.man.ac.uk/~jb/phys30101 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 4.4 Measurement of a spin component Magnetic moments (a) due to orbital angular momentum r e Electron in orbit produces a magnetic field (like bar magnet) and therefore has a magnetic dipole moment: μl = gl l μB μl = gl l μB μB = eħ/2me (the Bohr magneton) gl = -1 (gyromagnetic ratio or g-factor) l= 1, 2, 3… 4.4 Measurement of a spin component Magnetic moments (a) due to orbital angular momentum Electron in orbit produces a magnetic field (like bar magnet) and therefore has a magnetic dipole moment: r e μl = gl l μB μl = gl l μB μB = eħ/2me (the Bohr magneton) gl = -1 (gyromagnetic ratio or g-factor) l= 1, 2, 3… (b) Due to intrinsic spin μs = gs s μB gs = -2 exactly from Dirac theory gs = -2.0023192 (inc. QED corrections) μs = gs s μB s=½ thus μs = - μB is a good approximation Nuclear magnetic moments (10-3 smaller than atomic moments) Proton s = ½ μp = g s s μN Nuclear magneton μN Neutron s = ½ = eħ/2mp gs = +5.58 (not a Dirac point particle but must have substructure (quarks) μn = g s s μN gs = - 3.83 (also not a Dirac point particle. Quark model wavefunctions can explain moment) The Stern-Gerlach apparatus Rae, Eq. (5.30) 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 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. Coupling two angular momenta When M (= m1 + m2) is a constant of motion, m1 and m2 are not well defined S L We shall try and follow this convention: Capitals J, L, S indicate angular momentum vectors with magnitudes that can be expressed in units of ħ: L2 = l ( l + 1) ħ2 Lower case j, l, s indicate quantum numbers that are integer or half-integer: l = 0, 1, 2, 3… s = 1/2 j = 1/2 , 3/2, 5/2 Lower case vectors j, l, s indicate vectors whose components along a quantization axis are integer or halfinteger values (ie not in units of ħ).