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: π π π µ + µ + e + ν ν ν e µ µ + γ 99.988%

Units We use Gaussian units, thank you, Prof. Griffiths! q1q2 Yes Gaussian : F =, α = 2 r 1 q1q2 No SI : F =, 2 4πε r No HL : No 0 2 e c 1 137 1 q1q2 F =, 2 4π r = c = 1 or atomic units ( = e = 1)

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!

The Standard Model

Composite particles: it s like Greek to me

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!

The Universe today: little do we know!

Nuclear Physics Atomic Physics Particle Physics CM Physics Cosmology

Particle colliders: the tools of discovery PDG collider table CERN LHC video

Particle detectors: the tools of discovery Atlas detector: assembly First Z e + e - event at Atlas

Feynman diagrams

Feynman diagrams Professor Oleg Sushkov s notes, pp. 36-42: http://www.phys.unsw.edu.au/phys3050/pdf/particles_classification.pdf

Feynman diagrams Oleg Sushkov

Running coupling constants

The atmospheric muon paradox Mean lifetime: = 2.19703(4) 10 6 s c 6 10 4 cm = 600 m How do muons reach sea level? Relativistic time dilation

Lorentz transformations

Lorentz transformations: adding velocities

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

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)

Lorentz transformations: seen as hyperbolic rotations Rapidities: x moving α t stationary

Symmetries, groups, conservation laws

Symmetries, groups, conservation laws Symmetry: operation that leaves system unchanged Full set of symmetries for a given system Elements commute Abelian group Translations abelian; rotations nonabelian GROUP Physical groups can be represented by groups of matrices U(n) n n unitary matrices: SU(n) determinant equal 1 Real unitary matrices: O(n) 1 ~ U = U * SO(n) all rotations in space of n dimensions ~ O SO(3) the usual rotations (angular-momentum conservation) O 1 =

Angular Momentum First, a reminder from Atomic Physics

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 E n 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 (2l+ 1) = n = n 2 l= 0 2 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 l z and φ (azimuthal angle of the e): This is a bit more complex than (*) because φ is cyclic With definite l z, cosϕ = 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 2 L = Iω ~ mr ω Presumably, we want ω finite The surface of the object has linear velocity ~ ωr < c If we have L ~, Eqs. (1,2) r > = c 3.9 10 mc Experiment: electron is pointlike down to ~ 10-18 cm 11 (1) (2) 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! 45

Electron spin and fine structure (cont d) Both orbital angular momentum and spin have associated magnetic moments μ l and μ s Classical estimate of μ l : 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 μ s 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 m l and m s are no longer eigenstates States with fixed j, m j are eigenstates Total angular momentum is a constant of motion of an isolated system m j 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, j z 2 j = j = 0; j = j( j+ 1) x We can draw all of this as (j=3/2) y m j = 3/2 m j = 1/2 52

Vector model of the atom (cont d) These pictures are nice, but NOT problem-free Consider maximum-projection state m j = j m j = 3/2 Q: What is the maximal value of j x or j y 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,m l,s,m s 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!

Vector Model Example: a two-electron atom (He) Quantum numbers: J, m J l 1, l 2, L, S good no restrictions for isolated atoms good in LS coupling m l, m s, m L, m S not good =superpositions Precession rate hierarchy: l 1, l 2 around L and s 1, s 2 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

Flavor Symmetry Protons and neutrons a close in mass n is 1.3 MeV (out of 940 MeV) heavier than p Coulomb repulsion should make p heavier Isospin: p = Not in real space! No 1 0 Never mind terminology: isotopic, isobaric Strong interactions are invariant w.r.t. isospin projection n = 0 1

Flavor Symmetry Nucleons are isodoublet Pions are isotriplet: + π = 11 π 0 = 10 π = 1 1

Oleg Sushkov:

Oleg Sushkov: + π = 11 0 π = 1 0 π = 1 1