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Physics 129, Fall 2010; Prof. D. Budker Some introductory thoughts Reductionists’ science Identical particles are truly so (bosons, fermions) We will be using (relativistic) QM where initial conditions do not uniquely define outcome: 99.988% e e Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Units We use Gaussian units, thank you, Prof. Griffiths! 2 q1q2 e 1 Yes Gaussian : F 2 , r c 137 1 q1q2 No SI : F , 2 40 r No HL : No 1 q1q2 F , 2 4 r c 1 or atomic units ( e 1) Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Useful resource: Particle Data Group: http://pdg.lbl.gov/ The Particle Data Group is an international collaboration charged with summarizing Particle Physics, as well as related areas of Cosmology and Astrophysics. In 2008, the PDG consisted of 170 authors from 108 institutions in 20 countries. Order your free Particle Data Booklet ! Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The Standard Model Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The Standard Model Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The Standard Model Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The Standard Model Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The Standard Model Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Composite particles: it’s like Greek to me Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html In the beginning… First 4 chapters in Griffiths --- self review We will cover highlights in class Homework is essential! Physics Department colloquia and webcasts Watch Frank Wilczek’s Oppenheimer lecture Take advantage of being at Berkeley! Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The Universe today: little do we know! Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Nuclear Physics Atomic Physics Particle Physics CM Physics Cosmology Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Particle colliders: the tools of discovery PDG collider table CERN LHC video Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Particle detectors: the tools of discovery Atlas detector: assembly First Z e+e- event at Atlas Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Feynman diagrams Professor Oleg Sushkov’s notes, pp. 36-42: http://www.phys.unsw.edu.au/PHYS3050/pdf/Particles_classification.pdf Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Feynman diagrams Oleg Sushkov Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Feynman diagrams Oleg Sushkov Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Feynman diagrams Oleg Sushkov Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Feynman diagrams Oleg Sushkov Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Running coupling constants Renormalization……unification? * No hope of colliders at 1014 GeV ! need to learn to be smart! Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html The atmospheric muon “paradox” Mean lifetime: c = 2.19703(4)×10−6 s 6×104 cm = 600 m How do muons reach sea level? Relativistic time dilation Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Lorentz transformations Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Lorentz transformations: adding velocities Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html By the way… If we fire photons heads on, what is their relative speed? Moving shadows, scissors,… Garbage (IMHO): superluminal tunneling Confusing terminology (IMHO): “fast light” Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Lorentz transformations: Griffiths’ 3 things to remember • Moving clocks are slower (by a factor of > 1) • Moving sticks are shorter (by a factor of > 1) • Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Lorentz transformations: seen as hyperbolic rotations Rapidities: x moving α t stationary Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Symmetries, groups, conservation laws Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Symmetries, groups, conservation laws • Symmetry: operation that leaves system “unchanged” • Full set of symmetries for a given system • Elements commute GROUP Abelian group • Translations – abelian; rotations – nonabelian • Physical groups – can be represented by groups of matrices • U(n) – n n unitary matrices: ~* U U 1 • SU(n) – determinant equal 1 • Real unitary matrices: O(n) ~ O O 1 • SO(n) – all rotations in space of n dimensions • SO(3) – the usual rotations (angular-momentum conservation) Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Angular Momentum First, a reminder from Atomic Physics Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Angular momentum of the electron in the hydrogen atom Orbital-angular-momentum quantum number l = 0,1,2,… This can be obtained, e.g., from the Schrödinger Eqn., or straight from QM commutation relations The Bohr model: classical orbits quantized by requiring angular momentum to be integer multiple of There is kinetic energy associated with orbital motion an upper bound on l for a given value of En Turns out: l = 0,1,2, …, n-1 35 Angular momentum of the electron in the hydrogen atom (cont’d) In classical physics, to fully specify orbital angular momentum, one needs two more parameters (e.g., two angles) in addition to the magnitude In QM, if we know projection on one axis (quantization axis), projections on other two axes are uncertain Choosing z as quantization axis: Note: this is reasonable as we expect projection magnitude not to exceed 36 Angular momentum of the electron in the hydrogen atom (cont’d) m – magnetic quantum number because B-field can be used to define quantization axis Can also define the axis with E (static or oscillating), other fields (e.g., gravitational), or nothing Let’s count states: m = -l,…,l i. e. 2l+1 states l = 0,…,n-1 n 1 1 2( n 1) 1 2 (2 l 1) n n 2 l 0 37 Angular momentum of the electron in the hydrogen atom (cont’d) Degeneracy w.r.t. m expected from isotropy of space Degeneracy w.r.t. l, in contrast, is a special feature of 1/r (Coulomb) potential 38 Angular momentum of the electron in the hydrogen atom (cont’d) How can one understand restrictions that QM puts on measurements of angular-momentum components ? The basic QM uncertainty relation leads to (and permutations) We can also write a generalized uncertainty relation between lz and φ (azimuthal angle of the e): This is a bit more complex than (*) because φ is cyclic With definite lz , cos 0 (*) 39 Wavefunctions of the H atom A specific wavefunction is labeled with n l m : In polar coordinates : i.e. separation of radial and angular parts Spherical functions (Harmonics) Further separation: 40 Wavefunctions of the H atom (cont’d) Legendre Polynomials 41 Electron spin and fine structure Experiment: electron has intrinsic angular momentum -spin (quantum number s) It is tempting to think of the spin classically as a spinning object. This might be useful, but to a point L I ~ mr 2 (1) Presumably , we want finite The surface of the object has linear vel ocity ~ ωr c (2) If we have L ~ , Eqs. (1,2) r c 3.9 10 11 cm mc Experiment: electron is pointlike down to ~ 10-18 cm 42 Electron spin and fine structure (cont’d) Another issue for classical picture: it takes a 4π rotation to bring a half-integer spin to its original state. Amazingly, this does happen in classical world: from Feynman's 1986 Dirac Memorial Lecture (Elementary Particles and the Laws of Physics, CUP 1987) 43 Electron spin and fine structure (cont’d) Another amusing classical picture: spin angular momentum comes from the electromagnetic field of the electron: This leads to electron size Experiment: electron is pointlike down to ~ 10-18 cm 44 Electron spin and fine structure (cont’d) s=1/2 “Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is >/2 ! The square of the projection is always 1/4 45 Electron spin and fine structure (cont’d) Both orbital angular momentum and spin have associated magnetic moments μl and μs Classical estimate of μl : current loop For orbit of radius r, speed p/m, revolution rate is Gyromagnetic ratio 46 Electron spin and fine structure (cont’d) Bohr magneton In analogy, there is also spin magnetic moment : 47 Electron spin and fine structure (cont’d) The factor 2 is important ! Dirac equation for spin-1/2 predicts exactly 2 QED predicts deviations from 2 due to vacuum fluctuations of the E/M field One of the most precisely measured physical constants: 2=2 1:001 159 652 180 73 28 [0.28 ppt] Prof. G. Gabrielse, Harvard 48 Electron spin and fine structure (cont’d) 49 Electron spin and fine structure (cont’d) When both l and s are present, these are not conserved separately This is like planetary spin and orbital motion On a short time scale, conservation of individual angular momenta can be a good approximation l and s are coupled via spin-orbit interaction: interaction of the motional magnetic field in the electron’s frame with μs Energy shift depends on relative orientation of l and s, i.e., on 50 Electron spin and fine structure (cont’d) QM parlance: states with fixed ml and ms are no longer eigenstates States with fixed j, mj are eigenstates Total angular momentum is a constant of motion of an isolated system |mj| j If we add l and s, j ≥ |l-s| ; j l+s s=1/2 j = l ½ for l > 0 or j = ½ for l = 0 51 Vector model of the atom Some people really need pictures… Recall: for a state with given j, jz jx j y 0; j2 = j ( j 1) We can draw all of this as (j=3/2) mj = 3/2 mj = 1/2 52 Vector model of the atom (cont’d) These pictures are nice, but NOT problem-free Consider maximum-projection state mj = j mj = 3/2 Q: What is the maximal value of jx or jy that can be measured ? A: that might be inferred from the picture is wrong… 53 Vector model of the atom (cont’d) So how do we draw angular momenta and coupling ? Maybe as a vector of expectation values, e.g., Simple Has well defined QM meaning ? BUT Boring Non-illuminating Or stick with the cones ? Complicated Still wrong… 54 Vector model of the atom (cont’d) A compromise : j is stationary l , s precess around j What is the precession frequency? Stationary state – quantum numbers do not change Say we prepare a state with fixed quantum numbers |l,ml,s,ms This is NOT an eigenstate but a coherent superposition of eigenstates, each evolving as Precession Quantum Beats l , s precess around j with freq. = fine-structure splitting 55 Angular Momentum addition Q: q + anti-q = meson; What is the meson’s spin? A: 0 = ½ - ½ pseudoscalar mesons (π, K, 1 = ½ + ½ vector mesons (ρ, , , …) Can add 3 and more! Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html ’, …) Vector Model Example: a two-electron atom (He) Quantum numbers: “good” no restrictions for isolated atoms l1, l2 , L, S “good” in LS coupling ml , ms , mL , mS “not good”=superpositions J, mJ “Precession” rate hierarchy: l1, l2 around L and s1, s2 around S: residual Coulomb interaction (term splitting -- fast) L and S around J (fine-structure splitting -- slow) 57 jj and intermediate coupling schemes Sometimes (for example, in heavy atoms), spin-orbit interaction > residual Coulomb LS coupling To find alternative, step back to central-field approximation Once again, we have energies that only depend on electronic configuration; lift approximations one at a time Since spin-orbit is larger, include it first 58 Angular Momentum addition Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Flavor Symmetry Protons and neutrons are close in mass n is 1.3 MeV (out of 940 MeV) heavier than p Coulomb repulsion should make p heavier Isospin: 1 0 p n 0 1 Not in real space! No Never mind terminology: isotopic, isobaric Strong interactions are invariant w.r.t. isospin “projection” Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Flavor Symmetry Nucleons are isodoublet Pions are isotriplet: 11 0 10 1 1 Q: Does the whole thing seem a bit crazy? It works, somehow… Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Oleg Sushkov: Redundant slide Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Oleg Sushkov: Redundant slide 11 0 10 Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html 1 1 Proton and neutron properties Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html Proton and neutron properties Physics 129, Fall 2010, Prof. D. Budker; http://budker.berkeley.edu/Physics129_2010/Phys129.html