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: Physics 129, Fall 2010, Prof. D. Budker;

Units  We use Gaussian units, thank you, Prof. Griffiths! Physics 129, Fall 2010, Prof. D. Budker;

Useful resource:  Particle Data Group:  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 ! Order your free Particle Data Booklet ! Physics 129, Fall 2010, Prof. D. Budker;

The Standard Model Physics 129, Fall 2010, Prof. D. Budker;

The Standard Model Physics 129, Fall 2010, Prof. D. Budker;

The Standard Model Physics 129, Fall 2010, Prof. D. Budker;

The Standard Model Physics 129, Fall 2010, Prof. D. Budker;

The Standard Model Physics 129, Fall 2010, Prof. D. Budker;

Composite particles: it’s like Greek to me Physics 129, Fall 2010, Prof. D. Budker;

In the beginning…  First 4 chapters in Griffiths --- self review  We will cover highlights in class  Homework is essential!  Physics Department colloquia and webcastswebcasts  Watch Frank Wilczek’s Oppenheimer lectureFrank Wilczek’s Oppenheimer lecture  Take advantage of being at Berkeley! Physics 129, Fall 2010, Prof. D. Budker;

The Universe today: little do we know!

Physics 129, Fall 2010, Prof. D. Budker; Particle Physics Atomic Physics Cosmology Nuclear Physics CM Physics

Particle colliders: the tools of discovery Physics 129, Fall 2010, Prof. D. Budker; CERN LHC video PDG collider table

Particle detectors: the tools of discovery Physics 129, Fall 2010, Prof. D. Budker; Atlas detector: assembly First Z  e + e - event at Atlas

Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker;

Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker; Professor Oleg Sushkov’s notes, pp Professor Oleg Sushkov’s notes, pp :

Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker; Oleg Sushkov

Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker; Oleg Sushkov

Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker; Oleg Sushkov

Feynman diagrams Physics 129, Fall 2010, Prof. D. Budker; Oleg Sushkov

Running coupling constants Physics 129, Fall 2010, Prof. D. Budker; Renormalization……unification? * No hope of colliders at GeV !  need to learn to be smart!

The atmospheric muon “paradox” Physics 129, Fall 2010, Prof. D. Budker; Mean lifetimeMean lifetime:  = (4)×10 −6 s c   6×10 4 cm = 600 m How do muons reach sea level?  Relativistic time dilation

Lorentz transformations Physics 129, Fall 2010, Prof. D. Budker;

Lorentz transformations: adding velocities Physics 129, Fall 2010, Prof. D. Budker;

By the way… Physics 129, Fall 2010, Prof. D. Budker;  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 Physics 129, Fall 2010, Prof. D. Budker; Moving clocks are slower (by a factor of  > 1) Moving sticks are shorter (by a factor of  > 1)

Lorentz transformations: seen as hyperbolic rotations Physics 129, Fall 2010, Prof. D. Budker; Rapidities: t x α stationary moving

Symmetries, groups, conservation laws Physics 129, Fall 2010, Prof. D. Budker;

Symmetries, groups, conservation laws Physics 129, Fall 2010, Prof. D. Budker; Symmetry: operation that leaves system “unchanged” Full set of symmetries for a given system  GROUP Elements commute  Abelian group Translations – abelian; rotations – nonabelian 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) SO(n) – all rotations in space of n dimensions SO(3) – the usual rotations (angular-momentum conservation)

Angular Momentum  First, a reminder from Atomic Physics Physics 129, Fall 2010, Prof. D. Budker;

35 Angular momentum of the electron in the hydrogen atom Orbital-angular-momentum quantum number l = 0,1,2,… 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 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  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 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 Turns out: l = 0,1,2, …, n-1

36 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 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 In QM, if we know projection on one axis (quantization axis), projections on other two axes are uncertain Choosing z as quantization axis: Choosing z as quantization axis: Note: this is reasonable as we expect projection magnitude not to exceed Note: this is reasonable as we expect projection magnitude not to exceed

37 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 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 Can also define the axis with E (static or oscillating), other fields (e.g., gravitational), or nothing Let’s count states: Let’s count states: m = -l,…,l i. e. 2l+1 states m = -l,…,l i. e. 2l+1 states l = 0,…,n-1  l = 0,…,n-1 

38 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. m expected from isotropy of space Degeneracy w.r.t. l, in contrast, is a special feature of 1/r (Coulomb) potential Degeneracy w.r.t. l, in contrast, is a special feature of 1/r (Coulomb) potential

39 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 ? How can one understand restrictions that QM puts on measurements of angular-momentum components ? The basic QM uncertainty relation (*) leads to (and permutations) The basic QM uncertainty relation (*) leads to (and permutations) We can also write a generalized uncertainty relation 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 This is a bit more complex than (*) because φ is cyclic With definite l z, With definite l z,

40 Wavefunctions of the H atom A specific wavefunction is labeled with n l m : A specific wavefunction is labeled with n l m : In polar coordinates : In polar coordinates : i.e. separation of radial and angular parts Further separation: Further separation: Spherical functions (Harmonics)

41 Wavefunctions of the H atom (cont’d) Legendre Polynomials

42 Electron spin and fine structure Experiment: electron has intrinsic angular momentum -- spin (quantum number s) 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 It is tempting to think of the spin classically as a spinning object. This might be useful, but to a point Experiment: electron is pointlike down to ~ cm

43 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: 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)

44 Electron spin and fine structure (cont’d) Another amusing classical picture: spin angular momentum comes from the electromagnetic field of the electron: Another amusing classical picture: spin angular momentum comes from the electromagnetic field of the electron: This leads to electron size This leads to electron size Experiment: electron is pointlike down to ~ cm

45 Electron spin and fine structure (cont’d) s=1/2  s=1/2  “Spin up” and “down” should be used with understanding that the length (modulus) of the spin vector is >  /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 The square of the projection is always 1/4

46 Electron spin and fine structure (cont’d) Both orbital angular momentum and spin have associated magnetic moments μ l and μ s Both orbital angular momentum and spin have associated magnetic moments μ l and μ s Classical estimate of μ l : current loop Classical estimate of μ l : current loop For orbit of radius r, speed p/m, revolution rate is For orbit of radius r, speed p/m, revolution rate is Gyromagnetic ratio

47 Electron spin and fine structure (cont’d) In analogy, there is also spin magnetic moment : In analogy, there is also spin magnetic moment : Bohr magneton

48 Electron spin and fine structure (cont’d) The factor  2 is important ! The factor  2 is important ! Dirac equation for spin-1/2 predicts exactly 2 Dirac equation for spin-1/2 predicts exactly 2 QED predicts deviations from 2 due to vacuum fluctuations of the E/M field 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: [0.28 ppt] One of the most precisely measured physical constants:  2=2  1: [0.28 ppt] Prof. G. Gabrielse, Harvard

49 Electron spin and fine structure (cont’d)

50 Electron spin and fine structure (cont’d) When both l and s are present, these are not conserved separately When both l and s are present, these are not conserved separately This is like planetary spin and orbital motion This is like planetary spin and orbital motion On a short time scale, conservation of individual angular momenta can be a good approximation 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 l and s are coupled via spin-orbit interaction: interaction of the motional magnetic field in the electron’s frame with μ s l and s, i.e., on Energy shift depends on relative orientation of l and s, i.e., on

51 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

52 Vector model of the atom Some people really need pictures… Recall: for a state with given j, j z We can draw all of this as (j=3/2) m j = 3/2m j = 1/2

53 Vector model of the atom (cont’d) These pictures are nice, but NOT problem-free Consider maximum-projection state m j = j 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… m j = 3/2

54 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…

55 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

Angular Momentum addition Physics 129, Fall 2010, Prof. D. Budker;  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!

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

58 jj and intermediate coupling schemes Sometimes (for example, in heavy atoms), spin-orbit interaction > residual Coulomb  LS coupling Sometimes (for example, in heavy atoms), spin-orbit interaction > residual Coulomb  LS coupling To find alternative, step back to central-field approximation 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 

Angular Momentum addition Physics 129, Fall 2010, Prof. D. Budker;

Flavor Symmetry Physics 129, Fall 2010, Prof. D. Budker;  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:  Not in real space!  No   Never mind terminology: isotopic, isobaric  Strong interactions are invariant w.r.t. isospin “projection”

Flavor Symmetry Physics 129, Fall 2010, Prof. D. Budker;  Nucleons are isodoublet  Pions are isotriplet:  Q: Does the whole thing seem a bit crazy?  It works, somehow…

Physics 129, Fall 2010, Prof. D. Budker; Oleg Sushkov: Redundant slide

Physics 129, Fall 2010, Prof. D. Budker; Oleg Sushkov: Redundant slide

Physics 129, Fall 2010, Prof. D. Budker; Proton and neutron properties

Physics 129, Fall 2010, Prof. D. Budker; Proton and neutron properties