The tentative schedule of lectures for the semester with links to posted electronic versions of my notes can be found at:

Introduction to Elementary Particles, David Griffiths, Harper & Row (1987). Introduction to High Energy Physics, Donald Perkins, Addison-Wesley Publishing. The Fundamental Particles and Their Interactions, 1st Edition, William B. Rolnick

Two waves of slightly different wavelength and frequency produce beats.  x   x x 1k1k k = 2  NOTE: The spatial distribution depends on the particular frequencies involved

Many waves of slightly different wavelength can produce “wave packets.”

Adding together many frequencies that are bunched closely together …better yet… integrating over a range of frequencies forms a tightly defined, concentrated “wave packet” A staccato blast from a whistle cannot be formed by a single pure frequency but a composite of many frequencies close to the average (note) you recognize You can try building wave packets at

The broader the spectrum of frequencies (or wave number) …the shorter the wave packet! The narrower the spectrum of frequencies (or wave number) …the longer the wave packet!

Fourier Transforms Generalization of ordinary “Fourier expansion” or “Fourier series” Note how this pairs “canonically conjugate” variables  and t. Whose product must be dimensionless (otherwise e  i  t makes no sense!)

Conjugate variables time & frequency: t,  What about coordinate position & ???? r or x inverse distance?? wave number,  In fact through the deBroglie relation, you can write:

x0 x0 For a well-localized particle (i.e., one with a precisely known position at x = x 0 ) we could write: Dirac  -function a near discontinuous spike at x=x 0, (essentially zero everywhere except x=x 0 ) x0 x0 with such that f(x)≈ f(x 0 ), ≈constant over x  x, x+  x xx 1x1x

For a well-localized particle (i.e., one with a precisely known position at x = x 0 ) we could write: In Quantum Mechanics we learn that the spatial wave function  ( x ) can be complemented by the momentum spectrum of the state, found through the Fourier transform: Here that’s Notice that the probability of measuring any single momentum value, p, is: What’s THAT mean? The probability is CONSTANT – equal for ALL momenta! All momenta equally likely! The isolated, perfectly localized single packet must be comprised of an infinite range of momenta!

(k)(k) (x)(x) k0k0 (x)(x) (k)(k) k0k0 Remember: …and, recall, even the most general  whether confined by some potential OR free actually has some spatial spread within some range of boundaries!

Fourier transforms do allow an explicit “closed” analytic form for the Dirac delta function

Area within  1  68.26%  1.28  80.00%  1.64  90.00%  1.96  95.00%  2  95.44%  2.58  99.00%  3  99.46%  4  99.99% -2  -1  +1  +2   Let’s assume a wave packet tailored to be something like a Gaussian (or “Normal”) distribution A single “damped” pulse bounded tightly within a few  of its mean postion, μ.

For well-behaved (continuous) functions (bounded at infiinity) like f(x)=e -x 2 /2  2 Starting from: f(x)f(x) g'(x)g'(x)g(x)= e +ikx ikik f(x) is bounded oscillates in the complex plane over-all amplitude is damped at ±  we can integrate this “by parts”

Similarly, starting from:

And so, specifically for a normal distribution: f(x)=e  x 2 /2  2 differentiating: from the relation just derived: Let’s Fourier transform THIS statement i.e., apply:on both sides! 1  2  F'(k)e -ikx dk ~ ~ ~ e i(k-k)x dx ~ 1 2   (k – k) ~

e i(k-k)x dx ~ 1 2   (k – k) ~ selecting out k=k ~ rewriting as: 0 k 0 k dk' ' ' '

Fourier transforms of one another Gaussian distribution about the origin Now, since: we expect: Both are of the form of a Gaussian!  x    k  1/ 

 x  k  1 or giving physical interpretation to the new variable  x  p x  h

 x  k ~ 2   t  ~ 2   x  p ~ h  t  E ~ h

To be charged : means the particle is capable of emitting and absorbing photons What’s the ground state or zero-point energy of a system? e  e  harmonic oscillator: ½h

 x  p ~ h  t  E ~ h The virtual photons in the sea that surround every charged particle, are wavepackets centered at the origin (source of charge) If these describe/map out the electrostatic potential and relate available momentum to be transferred to the distance from the field’s source consider the extremes:  x  0  x   other charges may be exposed to the full spectrum of possible momenta only vanishingly small momentum transfers are possible

  1912

Henri Becquerel ( ) received the 1903 Nobel Prize in Physics for the discovery of natural radioactivity. Wrapped photographic plate showed clear silhouettes, when developed, of the uranium salt samples stored atop it While studying the photographic images of various fluorescent & phosphorescent materials, Becquerel finds potassium-uranyl sulfate spontaneously emits radiation capable of penetrating  thick opaque black paper  aluminum plates  copper plates Exhibited by all known compounds of uranium (phosphorescent or not) and metallic uranium itself.

1898 Marie Curie discovers thorium ( 90 Th) Together Pierre and Marie Curie discover polonium ( 84 Po) and radium ( 88 Ra) 1899 Ernest Rutherford identifies 2 distinct kinds of rays emitted by uranium  - highly ionizing, but completely absorbed by cm aluminum foil or a few cm of air  - less ionizing, but penetrate many meters of air or up to a cm of aluminum P. Villard finds in addition to  rays, radium emits  - the least ionizing, but capable of penetrating many cm of lead, several feet of concrete

B-field points into page Studying the deflection of these rays in magnetic fields, Becquerel and the Curies establish  rays to be charged particles

Using the procedure developed by J.J. Thomson in 1887 Becquerel determined the ratio of charge q to mass m for  : q/m = 1.76×10 11 coulombs/kilogram identical to the electron!  : q/m = 4.8×10 7 coulombs/kilogram 4000 times smaller!

Discharge Tube Thin-walled (0.01 mm) glass tube to vacuum pump & Mercury supply Radium or Radon gas Noting helium gas often found trapped in samples of radioactive minerals, Rutherford speculated that  particles might be doubly ionized Helium atoms (He ++ ) Rutherford and T.D.Royds develop their “alpha mousetrap” to collect alpha particles and show this yields a gas with the spectral emission lines of helium! Mercury

Status of particle physics early 20th century Electron J.J.Thomson 1898 nucleus (  proton ) Ernest Rutherford  Henri Becquerel 1896 Ernest Rutherford 1899   P. Villard 1900 X-rays Wilhelm Roentgen 1895

1900 Charles T. R. Wilson’s ionization chamber Electroscopes eventually discharge even when all known causes are removed, i.e., even when electroscopes are sealed airtight flushed with dry, dust-free filtered air far removed from any radioactive samples shielded with 2 inches of lead! seemed to indicate an unknown radiation with greater penetrability than x-rays or radioactive  rays Speculating they might be extraterrestrial, Wilson ran underground tests at night in the Scottish railway, but observed no change in the discharging rate.

1909 Jesuit priest, Father Thomas Wulf, improved the ionization chamber with a design planned specifically for high altitude balloon flights. A taut wire pair replaced the gold leaf. This basic design became the pocket dosimeter carried to record one’s total exposure to ionizing radiation. 0

1909 Taking his ionization chamber first to the top of the Eiffel Tower (275 m) Wulf observed a 64% drop in the discharge rate. Familiar with the penetrability of radioactive  rays, Wulf expected any ionizing effects due to natural radiation from the ground, would have been heavily absorbed by the “shielding” layers of air.

light produces spots of submicroscopic silver grains a fast charged particle can leave a trail of Ag grains 1/1000 mm (1/25000 in) diameter grains small singly charged particles - thin discontinuous wiggles only single grains thick heavy, multiply-charged particles - thick, straight tracks 1930s plates coated with thick photographic emulsions (gelatins carrying silver bromide crystals) carried up mountains or in balloons clearly trace cosmic ray tracks through their depth when developed

November 1935 Eastman Kodak plates carried aboard Explorer II’s record altitude (72,395 ft) manned flight into the stratosphere

50  m Cosmic ray strikes a nucleus within a layer of photographic emulsion 1937 Marietta Blau and Herta Wambacher report “stars” of tracks resulting from cosmic ray collisions with nuclei within the emulsion

Elastic collision

p p p p p p

1894 After weeks in the Ben Nevis Observatory, British Isles, Charles T. R. Wilsonbegins study of cloud formation a test chamber forces trapped moist air to expand supersaturated with water vapor condenses into a fine mist upon the dust particles in the air  each cycle carried dust that settled to the bottom  purer air required larger, more sudden expansion  observed small wispy trails of droplets forming without dust to condense on!

Tracks from an alpha source

1952 Donald A. Glaser invents the bubble chamber boiling begins at nucleation centers (impurities) in a volume of liquid along ion trails left by the passage of charged particles in a superheated liquid tiny bubbles form for ~10 msec before obscured by a rapid, agitated “rolling” boil hydrogen, deuterium, propane(C 3 H 6 ) or Freon(CF 3 Br) is stored as a liquid at its boiling point by external pressure (5-20 atm) super-heated by sudden expansion created by piston or diaphragm bright flash illumination and stereo cameras record 3 images through the depth of the chamber (~6  m resolution possible) a strong (2-3.5 tesla) magnetic field can identify the sign of a particle’s charge and its momentum (by the radius of its path) 1960 Glaser awarded the Nobel Prize for Physics

3.7m diameter Big European Bubble Chamber CERN (Geneva, Switzerland) Side View Top View

1936 Millikan’s group shows at earth’s surface cosmic ray showers are dominated by electrons, gammas, and X-particles capable of penetrating deep underground (to lake bottom and deep tunnel experiments) and yielding isolated single cloud chamber tracks

1937 Street and Stevenson 1938 Anderson and Neddermeyer determine X-particles are charged have 206× the electron’s mass decay to electrons with a mean lifetime of 2  sec sec

Schrödinger’s Equation Based on the constant (conserved) value of the Hamiltonian expression total energy  sum of KE + PE with the replacement of variables by “operators” As enormously powerful and successful as this equation is, what are its flaws? Its limitations?

We could attempt a RELATIVISTIC FORM of Schrödinger: What is the relativistic expression for energy? relativistic energy-momentum relation As you’ll appreciate LATER this simple form (devoid of spin factors) describes spin-less (scalar) bosons For m=0 this yields the homogeneous differential equation: Which you solved in E&M to find that wave equations for these fields were possible (electromagnetic radiation).

(1935) Hideki Yukawa saw the inhomogeneous equation as possibly descriptive of a scalar particle mediating SHORT-RANGE forces like the “strong” nuclear force between nucleons (ineffective much beyond the typical meter extent of a nucleus For a static potential drop and assuming a spherically symmetric potential, can cast this equation in the form: with a solution (you will verify for homework): where R= h mc

where R= h mc Let’s compare: to the potential of electromagnetic fields: with e -r/R =1 its like R  or m = 0! For a range something like m Yukawa hypothesized the existence of a new (spinless) boson with mc 2 ~ 100+ MeV. In 1947 the spin 0 pion was identified with a mass ~140 MeV/c 2

1947 Lattes, Muirhead, Occhialini and Powell observe pion decay  Cecil Powell ( 1947 ) Bristol University

C.F.Powell, P.H. Fowler, D.H.Perkins Nature 159, 694 (1947) Nature 163, 82 (1949)

Quantum Field Theory Not only is energy & momentum QUANTIZED (energy levels/orbitals) but like photons are quanta of electromagnetic energy, all particle states are the physical manifestation of quantum mechanical wave functions (fields). Not only does each atomic electron exist trapped within quantized energy levels or spin states, but its mass, its physical existence, is a quantum state of a matter field. ee the quanta of the em potential  virtual photons as opposed to observable photons These are not physical photons in orbitals about the electron. They are continuously and spontaneously being emitted/reabsorbed.

The Boson Propagator What is the momentum spectrum of Yukawa’s massive (spin 0) relativistic boson? Remember it was proposed in analogy to the E&M wave functions of a photon. What distribution of momentum (available to transfer) does a quantum wave packet of this potential field carry? q  r = qrcos  dV = r 2 d  sin  d  dr Integrating the angular part: 22 The more massive the mediating boson, the smaller this distribution…

Consistently ~600 microns (0.6 mm) 

pdg.lbl.gov/pdgmail

BraKet notation We generalize the definitions of vectors and inner products ("dot" products) to extend the formalism to functions (like QM wavefunctions) and differential operators. v = v x x + v y y + v z z   n v n n then the inner product is denoted by v  u = ^ ^^^ n vn unn vn un sometimes represented by row and column matrices: [ v x v y v z ] u x u y =[] u z v x u x + v y u y + v z u z Remember: n  m =  nm ^^

We most often think of "vectors" in ordinary 3-dim space, but can immediately and easily generalize to COMPLEX numbers: v  u =  n [ v x v y v z ] u x u y =[ ] u z n vn* unn vn* un v x * u x + v y * u y + v z * u z and by the requirement = < v | u > * we guarantee that the “dot product” is real transpose column into row and take complex conjugate ***

Every “vector” is a ket : | v 1 >| v 2 > including the unit “basis” vectors. We write: | v > =  n  | > and the scalar product by the symbol and the orthonormal condition on basis vectors can be stated as =  Now if we write | v 1 > =  C 1 n | n > and | v 2 > =  C 2 n |n> then “ we know ”: =  n C 2 n * C 1 n =  because of orthonormality =  m “bra” Cn nCn n v u m n mn  n,m C 2 m * C 1 n  m C 2 m *

So if we write| v > =  C n |n> =  n |n> =  n = {  n } = So what should this give? = ?? Remember: gives a single element 1 x 1 matrix but: | m > < n | gives a ??? C1nC1n |n> | v >| v > 1 |v>

 n |n><n| In the case of ordinary 3-dim vectors, this is a sum over the products: [ ] [ ] [ ] = =

ee Two important BASIC CONCEPTS The “coupling” of a fermion (fundamental constituent of matter) to a vector boson ( the carrier or intermediary of interactions ) Recognized symmetries are intimately related to CONSERVED quantities in nature which fix the QUANTUM numbers describing quantum states and help us characterize the basic, fundamental interactions between particles

Should the selected orientation of the x-axis matter? As far as the form of the equations of motion? (all derivable from a Lagrangian) As far as the predictions those equations make? Any calculable quantities/outcpome/results? Should the selected position of the coordinate origin matter? If it “doesn’t matter” then we have a symmetry: the x-axis can be rotated through any direction of 3-dimensional space or slid around to any arbitrary location and the basic form of the equations…and, more importantly, all the predictions of those equations are unaffected.

If a coordinate axis’ orientation or origin’s exact location “doesn’t matter” then it shouldn’t appear explicitly in the Lagrangian! EXAMPLE: TRANSLATION Moving every position (vector) in space by a fixed a (equivalent to “dropping the origin back” – a ) original description of position r –a–a r'r' new description of position or

For a system of particles: acted on only by CENTAL FORCES: function of separation no forces external to the system generalized momentum (for a system of particles, this is just the ordinary momentum) = for a system of particles T may depend on  q or  r but never explicitly on q i or r i

For a system of particlesacted on only by CENTAL FORCES: -F i a ^ net force on a system experiencing only internal forces guaranteed by the 3 rd Law to be 0 Momentum must be conserved along any direction the Lagrangian is invariant to translations in.