PHYS Quantum Mechanics PHYS Quantum Mechanics Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) These slides at: Lecture 17

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: μ l = g l l μ B μ B = eħ/2m e (the Bohr magneton) g l = -1 (gyromagnetic ratio or g-factor) l= 1, 2, 3… μ l = g l l μ B

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: μ l = g l l μ B μ B = eħ/2m e (the Bohr magneton) g l = -1 (gyromagnetic ratio or g-factor) l= 1, 2, 3… (b) Due to intrinsic spin μ s = g s s μ B g s = -2 exactly from Dirac theory g s = (inc. QED corrections) s = ½ thus μ s = - μ B is a good approximation μ s = g s s μ B μ l = g l l μ B

Nuclear magnetic moments (10 -3 smaller than atomic moments) Proton s = ½ μ p = g s s μ N Nuclear magneton μ N = eħ/2m p g s = (not a Dirac point particle but must have substructure (quarks) Neutron s = ½ μ n = g s s μ N g s = (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 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

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.

Coupling two angular momenta L S When M (= m 1 + m 2 ) is a constant of motion, m 1 and m 2 are not well defined

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: L 2 = l ( l + 1) ħ 2 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 half- integer values (ie not in units of ħ).