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