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QM Foundations ofQM Foundations of Particle Particle PhysicsPhysics
Chris Parkes April/May 2003
Hydrogen atomHydrogen atom Quantum numbers Electron intrinsic spin
Other atomsOther atoms More electrons! Pauli Exclusion Principle Periodic Table
EquationsEquations Towards QFT…Towards QFT… Klein-Gordon Dirac
AntiparticlesAntiparticles Discovery of the Positron
Relativistic Quantum MechanicsAtomic Structure1st Handout
Second Handout
http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html
References are to ‘Particle Physics’ - Martin&Shaw 2nd edition
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 Some alternative reasons to study this course!course!
2002 – neutrinos 1999 -- QFT1995 -- tau / neutrino 1992 -- particle detectors1990 -- Deep Inelastic Scattering 1988 -- muon neutrino1984 -- 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
Adding Relativity to QMAdding Relativity to QM
Free particle Em
2
2p Apply QM prescription ipt
iE
Get Schrdinger Equationdt
im
22
2Missing phenomena:Anti-particles, pair production, spin
Or non relativisticWhereas relativistically
m
pmvE
22
1 22
42222 cmcE p
22
2
2
2
1
mc
dtcKlein-Gordon Equation
Applying QM prescription again gives:
22
ckmc
KG is 2nd order in time
compton wavelength
mc
kc
A characteristic scale for relativity in QM
is called d’Albertian – four-vector differential operator version of del
SolutionsSolutions
2 solns not surprising – we started with a quadratic energy equation.
/).(),( EtiNet xpx (Same as non-relativistic)
With 24222 21
)( mccmcpE [show this]
But also satisfied by complex conjugate/).(*),( EtieNt xpx
With 24222 21
)( mccmcpE
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
Klein ParadoxKlein Paradox
Incident
ReflectedTransmitted
V0
Consider particles obeying KG eqn of mass m, charge q, hitting a potential barrier
LHS RHS // Re)( ipxipxex /')( xipTex
From considering continuity of wavefunction and derivative at boundary
pp
ppR
'
'
21
)( 4222 cmcpE 21
)'()( 4222 cmcpqVE o
pp
pT
'
2
This has some strange features! Due to p,E relationship e.g.• p’ is +ve imaginary, this is standard case for a large barrierHence 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 barrierp’<0 anti-particles travelling away from barrier .
Dirac EquationDirac EquationK-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:
),(),(),(
tixHt
ti x
x
With Hamiltonian H
First order in derivative of t, want first order in momentum()
3
1
2),(
i ii mcx
cit
tih
x
2mcciH pα First order momentum term + rest mass term
Ingeneously, he demanded the eqn squared match E2=p2+m2
12222 zyx gives
ijji and ji ji for
Can’t satisfy with numbers, but can with matrices
Dirac SpinorsDirac SpinorsSimplest matrices that fulfil commutation relations are 4x4, one representation is
10
01
10
011
00
000
0
0
i
ii
01
101
0
02 i
i
10
013
Plane wave solutions of the equation
)/.()(),( Etxpieput xWith four components
),(
),(
),(
),(
),(
4
3
2
1
t
t
t
t
t
x
x
x
x
x
There are four solutions• two with +ve Energy, two with –ve (anti-particles)•Spin +½ , Spin -½
PositronPositronKG 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
Anomalous magnetic momentAnomalous magnetic moment
BgmU BsRecall Schrodinger equation gives g=1
Apply potential to Dirac equation and look for term in S.B
Get g=2 For Dirac particles, fundamental spin ½ particles (electron,muon….)
meSD /pP meS /79.2whereas nn meS /91.1
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.
=(g-2)/2
Anti-particlesAnti-particlesResolving the problem of negative energy solutions
Why can’t the electron in a Hydrogen atom not drop below the ground state intoA negative state?
Dirac hole theoryDirac hole theoryDirac 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.
-mc2
E
mc2
etc..
etc.. E
mc2
etc..
etc.. E
mc2
etc..
etc..
vacuum electron positron
• removing a state with –E,-S,-p,-e•Leaves a ‘sea’ with +ve quantities!•Including +ve e charge
See 1.2.2Martin&Shaw
•Feynman et al.- +ve E states of a different particle