1. QM Foundations of Particle Physics QM Foundations of Particle Physics Chris Parkes April/May 2003  Hydrogen atom Quantum numbers Electron intrinsic spin  Other atoms More electrons! Pauli Exclusion Principle Periodic Table  Equations  Towards QFT… Klein-Gordon Dirac  Antiparticles Discovery of the Positron Relativistic Quantum Mechanics Atomic Structure 1 st Handout Second Handout http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html References are to ‘Particle Physics’ - Martin&Shaw 2 nd edition

2. LHC @ CERN 27 km long tunnel,100m underground French/Swiss Border near Geneva 1989 – 2000 Large Electron Positron collider (LEP), colliding beam synchotron 200 GeV 2007 onwards Large Hadron Collider (LHC), 14 TeV proton collider Some alternative reasons to study this course! 2002 – neutrinos1999 -- QFT 1995 -- tau / neutrino1992 -- particle detectors 1990 -- Deep Inelastic Scattering1988 -- muon neutrino 1984 -- W&Z bosons I want to understand why 5000 physicist worldwide are currently building the world’s largest machine! I want a nobel prize! Origin of mass (Higgs Boson) New Physics (e.g. Supersymmetry) Matter anti-matter asymmetry in Universe (CP Violation) Probably 2 PhD places in Glasgow for Oct. 2004 on this

3. Adding Relativity to QM Free particleApply QM prescription Get Schr  dinger Equation Missing phenomena: Anti-particles, pair production, spin Or non relativistic Whereas relativistically Klein-Gordon Equation Applying QM prescription again gives: KG is 2 nd order in time compton wavelength A characteristic scale for relativity in QM  is called d’Albertian – four-vector differential operator version of del

4. Solutions 2 solns not surprising – we started with a quadratic energy equation. (Same as non-relativistic) With[show this] But also satisfied by complex conjugate With But we seem to now have negative energy, a +Et term Particle with p,E Particle with -p,-E Or a particle with -p,+E and negative t. Negative t? a particle travelling backwards in time. Anti-particle can be considered as a particle travelling backwards in time. - we use this when labelling Feynman graphs Discuss features in terms of the Klein Paradox

5. Klein Paradox Incident Reflected Transmitted V0V0 Consider particles obeying KG eqn of mass m, charge q, hitting a potential barrier LHSRHS From considering continuity of wavefunction and derivative at boundary This has some strange features! Due to p,E relationship e.g. p’ is +ve imaginary, this is standard case for a large barrier Hence exp(ip’x) represents exp(–kx) decaying exponential, i.e. less likely to be behind boundary. Larger Barrier: p’ real, can choose p’–ve, then T,R >1! Enough energy for pair production ! Particle/anti-particle pairs emitted at barrier p’<0 anti-particles travelling away from barrier.

6. Dirac Equation K-G equation has introduced some properties we wanted but not spin. KG is the equation of spin 0 particles (bosons) Dirac is the equation of spin ½ fermions Again try an equation of the form: With Hamiltonian H First order in derivative of t, want first order in momentum(  ) First order momentum term + rest mass term Ingeneously, he demanded the eqn squared match E 2 =p 2 +m 2 gives and for Can’t satisfy with numbers, but can with matrices

7. Dirac Spinors Simplest matrices that fulfil commutation relations are 4x4, one representation is Plane wave solutions of the equation With four components There are four solutions two with +ve Energy, two with –ve (anti-particles) Spin +½, Spin -½

8. Positron KG as old as QM, originally dismissed. No spin 0 particles known. Pion was only discovered in 1948. Dirac equation of 1928 described known spin ½ electron. Also described an anti-particle – Dirac boldly postulated existence of positron Discovered by Anderson in 1933 using a cloud chamber (C.Wilson) Track curves due to magnetic field F=qvxB

9. Anomalous magnetic moment Recall Schrodinger equation gives g=1 Apply potential to Dirac equation and look for term in S.B Get g=2For Dirac particles, fundamental spin ½ particles (electron,muon….) whereas Measurement of proton magnetic moment was first indication that proton was not an elementary particle (1933) Also important for spin-orbit interaction and fine structure in atomic line splitting Magnetic moment of muon is measured in very precise experiments, looking at precession of spin for muons travelling in a circular ring.

10. =(g-2)/2

11. Anti-particles Resolving the problem of negative energy solutions Why can’t the electron in a Hydrogen atom not drop below the ground state into A negative state? Dirac hole theory Dirac hypothesis: the negative energy states are almost always filled, and pauli exclusion principle applies. A ‘sea’ of filled –ve states, no net spin or momentum. -mc 2 E mc 2 etc.. E mc 2 etc.. E mc 2 etc.. vacuum electron positron removing a state with –E,-S,-p,-e Leaves a ‘sea’ with +ve quantities! Including +ve e charge See 1.2.2 Martin&Shaw Feynman et al.- +ve E states of a different particle