... CONTINUATION ....
8. Quantum co-operation and superfluids
a. Laser light
+ Lasers from special property of wave motion
known as "coherence", light photons
act together in quantum co-operation.
+2 wave motions w/ phase differences.
Coherent wave if there is a definite phase difference.
Incoherent which don't show interference effects. ()
+Laser light remarkable since many dfferent
atoms is related in phase
+Laser()
+Need population inversaion pump energy, so stimulated
absorption, spontaneous emission of coherent photons.
+#INCL Fig 8.7 p112
+Lasing possible since photons allowed in same quantum state,
since they are "bosons".
+Pauli doesn't allow more than one to have same quantum number, but
photons prefer to be all together in same quantum state.
+Holography
b. Bose condensation and superfluid helium
+Pauli's exclusion principle applied to atomic electrons
mater-like particles (e- , p+ , n) obey it. No two identical
"fermions" can occupy same quantum sttate. So putting e- in a
box potential, e- cannot all go into lowest energy level.
But must fill quantized energy levels in pairs w/ opposite spoin.
So no two e- have same quantum number.
+For Bosons (radiation-like particles e.g. photons) -
behave differently. Prefer to be in same state.
+Einstein noted Bose particles (bosons) can all
condense down into lowest energy state. Particles in a box,
lowest energy state achieved by all of them occupying
lowest energy level. Einstein pointed
out from a certain temperature on, the molecules"condense"
without attractive forces.
+Bose condensation - liquid helium can be understood in Bose
condensation. How? He = fermion.
+4He contains even # of fermions - 2P= and 2n, 2e-. elements w/
even # of fermions can behave like bosons. So 4He can undergo
Bose condensation at low temperatures to show superfluid
behaviour. Elements w/ odd number of fermions always obey
Pauli Principle.
+Liquid helim - zero-point motion of helium
atoms - necessary quantum jiggling to satisfy
heisenberg uncertainty principle. Race to
liquify helium. 6k.
at 2k, boiling ceases liquid becomes still.
the viscosity (treacliness) drops by 1millions.
ability of liquid helium to creep along walls of
its container. Film acts like a siphons to allow
He to flow out.
+Result of helium atoms condensing into lowest energy state.
forming quantum superfluid.
+3He has odd number of fermions , but a 0.002K a new form
of Bose condensation is seen!!
At this temperature weak attractive forces between 2 3He atoms
sufficiently strong for pair to bind together act like a boson.
Pairs of 3He atoms can undergo a similar Bose condensation to that
of individual 4He atoms. similar mechanism is responsible for
superconductivity
c. Superconductivity
+Superconductors ()
+Superconduction levitation ()
+Cooper Pair ()
+Cooper pairs act like bosons, condense to form the superconducting
state. Since 0 momentum , Heisenberg -> position sphere is huge.
+BCS Theory ()
+Superconductivity applications ()
9. Feynman rules
a. Dirac and antiparticles
+ E^2 = (p^2*c^2) + (m^2*c^4)
E = (p^2) / (2*m)
+ Schrodinger needed a relativitstic wavefunction.
+Paul Dirac predicted antimatter.
+ E = +/- sqrt ( p^2*c^2 + m^2c^4 )
Thus an "empty" quantum box, w/ no positive
energy e- has a fully occupied "sea" of negative
energy electrons!
+electron-positron pair creation by a photon
qv electrons can absorb en from photon and jump to excited state.
+ E(psi) = ( -ialpha*del + betam)*Psi
Dirac equation for relativistic electron
+Proof : Construct accelerator w/ enough energy for proton-antiproton pair
via p + p -> p + p + p + !p
or annihilation e+ + e- -> photon + photon (annihilation energy)
+new feature of relativistic quantum mechanics, pair creation,
annihilation gives rise to possibility of transforming energy into matter.
So # of quantum particles can change.
+ Dirac relies on Pauli principle to prevent positive energy particles jumping
to a lower energy state, so "filled sea" trick doesn't work for bosons.
+Many-body quantum theory use particle creation and
annihilation from the outset., example of a quantum field theory.
+QFT describes interactions of electrons + photons = QED (quantum
electrodynamics)
+QED combines Maxwell's equations of electromagnetism, quantum mechanics and
relativity. Example()
>>
Spin of e- , spining e- like little magnet
strength of magnetic moment of e- calculated in QED.
(expressed in g-factor) for electron
gquantum = 2.0023193048 = gexperiment.
>>
b. Feynman diagrams and virtual particles
+
scatter e- back in time pair annihilation
| o \ \ | o
| / \ / o = o | / \ /
| / \ / (-)en e- moving back | / \ /
|/ o in time = positive en e+ | / o
+---------- moving fwd in time +--------------
scatter (-) energy pair creation
e- back in time to
ordinary e- fwd in
time
+virtual particle. w/ quantum tunneling
(del E)*(del t) ~= h (planck's constant)
so can borrow energy del E for a small time del t.
energy-time uncertainty relation.
So a particle need not always remain the same particle
can borrow enough energy to create another particle
or pair of particles which only exist for a short time.
called Virtual processes. and particles created = virtual particles.
+#INcl Fig 9.6, #INCL Fig #9.10
+Feynman diagrams can calculate probability amplitudes for scattering processes.
c. Zero-point motion and vacuum fluctuation
+The ground state of relativistic quantum field theory
(quantum foam) "empty space" is full of virtual particle-antiparticle pairs
+Vibrations of atoms in a crystal, vibrational waves setup
in the crystal have particle-like aspects, like photons.
Quantum lattice vibrations called phonons. If lattice
cooled down so no vibrational phonons excited, must still
be some zero-point motion of atoms.
+Photons = vibrations of EM field,
phonons quantum objects = vibration of crystal positions.
+Lamb Shift()
>>
Hydrogen tiny split in spectral line.
vacuum fluctuations of EM field cause e- in H atom
to jiggle. Excited e- undergo spontaneous emission
of a photon and jump to lower state. Transition = zero-point motion
of EM field.
>>
+vacuum force () - casimir effect
>>
all wavelength ok for EM field.
2 plates, EM field required to vanish on these plates
so zero-pt energy altered. So attractive force
between plates.
>>
d. Hawking radiation and black holes
+Hawking proposes whole range of masses can be formed during
early stages of the universe. Primordial black holes.
+Big Bang theory () - universe is expanding hubble redshift
>>
characteristic absorption lines of
elements are red shifted , evidence that the galaxies all moving
away from us and from each other. so expanding.
Hubbles law v = H*d
The further away from us a galaxy is , the faster it appears to be moving
away
>>
+Hawking also developed theory of particle creation near
primordial black holes. from tidal forces.
+Same tial forces cause particle-antiparticle pair to
become separated if they are created it he enormous
gravitational field of a primoridal black hole.
Possible for one of the pair to fall
into the hole while the other escapes into
the surroudnging space. The black hole
will thus appear to radiat particles, Hawking radiation.
miniblack hole will evaporate and then explode.
/*---------------------------------------------------------------------*/
. Weak photons and strong glue
a. The double-slit experiment revisited
b. The birth of particle physics
c. Weak photons and the Higgs vacuum
d. Quarks and gluons
e. Superconductors, magnetic monopoles and quark confinement
a.
+QED Quantum electrodynamics - combines classical EM, quantum mechanics
and relativity
+ WANT : (1) theory for weak force theory - natural radioactivity
(2) Strong force theory - hold nucleus together.
+Unifty EM + Weak force
+Theory for strong nuclear force - QCD
quantum chromodynamics. Theory of p+ and n in terms of
their quark constituents.
+All above theory are gauge invariant.
+(a) Fig 10.1 #INCL
Interference pattern for usual double-slit experiment w/ e-, e- arrive
individually at detector, reprsented as B/W circles. Pattern is
highest at central detector
(b) Interference pattern unchanged if a thin sheet of matter is
inserted between slits and detectors, electron waves coming from
both of the slits undergo the same change in phase. At
the dector the two waves will still add up to give a peak or dip
So "Global" Phase invariance since the pattern is unchanged -"Invariant"
provided the sheet crosses the entire region behind the slits.
(c) If a thin sheet of mtter is inserted behind only one of the
slits the pattern changes. Instead of the detector at A registering
a maximum, now records a dip. Pattern changed since matter has altered
the phase of only one of the electron waves. Shows "local" change in phase
does not leave the interference pattern "invariant"
So does not have local phase invariance
(d) The presence of a magnetic field will also cause the interference
pattern to change. Apart from the fact that we cannot say which slit the
electron passed through, you would expect this, since electrons are
deflected by magnetic fields.
(e) Famous Bohm-Aharanov experiment showed that there was a
shift in the interference pattern even if the magnetic field was shielded
so that the electron paths from each slit to the detector did not
pass through anymagnetic field! the shielded magnetic field
has been achieved in practice by using a long, thin EM coil, thinner
in diameter than a human hair.
(f) The pase shift caused by the insertion of a sheet of
matter behind one of the slits can be exactly compensated for by adjusting
the magnetic field. This shows that "local invariance"
can be achieved provided the magnetic field interacts with the
electron in just the right way.
This is the basic princple behind all "gauge" theories.
e.g. cats population are locally conserved
+Physics is intrigued with invariance principles.
Can we do better w/ global phase invariance?
Yes! Is there some way of fixing things up so we can
allow a local alteration of phase and yet still have an invariance?
Result is QED!
+10e 'Bohm Aharanov ' Effect1960
+10f used shielded magnet + slit-2 phase sheet of matter.
get local phases invariance, because locally change
phase w/ sheet of matter, but globally preserve
phase by having sheet of matter cancel effects of
shielded magnet effects (10e - bohm aharanov effects)
+QED theory that has interactions of electrons
and EM photons arranged to maintain local phase invariance
of the e- wavefunction.
This is called "gauge invariance" = "local phase invariance"
so QED is called a "gauge" theory = "phase " theory.
+How does this help find theory of weak || strong
interactions? reverse the argument. i.e. suppposed
didn't know how e- and photons interact. demand any theory w/
e- and must have local phase invariance. Forced to introduce B fields,
and invent QED! - -> This is "gauge principle"L
demand local phase invariance determines the nteractions of the theory.
b. birth of ....
1932 chadwick finds neutrons (thought all things made of e-,p+,n)
p+ and n called baryons (greek barys = heavy)
e- called lepton (greek word leptos = lightweight)
v (neutrino) is a lepton (little neutral one)
+Feynman diagrams show interactions of particles #INCLfig10.2
+In INCL #10.2 virtual photon is exchanged between e-
and a quark from the proton.
From uncertainty can borow delE for a time delt=h/delE
without spoiling energy conservation, multiply delt by velocity v
of the particle we can obtain estimate of the typical distance
such a particle can travel.
R = v (h/delE)
+Hideki yukawa predict existence of particles w/
mass between e- and proton. 1935
+Conversi,Pancini,Piccioni new particle didn't behave
like carrier of the strong force. instead of interacting strongly w/
nucleus, interacted more like an electron.
+1947 might be 2 new particles w/ about same mass. other was Yukawa's strong force carrier.
Intermediate mass particles called mesons "from mesos (middle gr)
name of heavy electrons called muons.
+Yukawa's discovery of meson allow better way to observe
collisions of high energy particles.
+1975 Martin Perl discovers tau-lepton
Yukawa's meson called pi-meson (pion)
++Double vee events are common in buble
chambers, a negativtly charged pion collides at A
with a proton of the hydrogen filling the chamber, the reaction
2 particles are produced a neutral K meson and a neutral
lambda baryon. The lambda decays at B
to a proton and a pion-,
the kaon at C to a pion+ and pion-
Puzzling - lambda and Kaon left to themselves, showed
marked reluctance to turn back into protons and pions.
production of pairs of strange particles via strong interaction
but decay of individual strange particles governed by weak interactions.
Now know strange particles posses a new type of charge,
in the strong interaction production reactions, the
initial and final states must always have the same strangeness. Kaon has strangeness
+1
and the lambda particle
strangeness -1.
so can't take place by fast strong interaction, but by slower weak
interactions of beta radioactivity.
+1960's Gell-Mann and Zweig introduce quarks.
Baryons from 3 quarks,
predict omega-
mesons of a quark and an antiquark
+50's60's found many short-lived excited states of the proton, neutron and pion.
can understand excited states of atoms and nuclei w/ quark systems.
+Baryons and mesons interact via strong nuclear force, = hadrons
(hadros gr = bulky)
c. weak photons and higgs vacuum.
1979 Nobel Glashow, Salam , Weinberg theory of
unified weak interaction between elementary particles
+predict W and Z bosons.
+How does unification of weak and EM interaction come about? what has this to do w/ gauge principle?
++->Yukawa argument about range of forces and this is
relation w/ mass of the virtual particle being exchanged.
Yukawa deduced the mass of the pion
from the observed range of nuclear forces.
leptons feel only the weak and EM forces.
The heavier a particle is, the more energy that has to
be borrowed to create it, and the shorter the distance
it can travel on "borrowed time"
+Energy E, Momentum p and mass m of
a particle moving at very high, relativistic speeds related by
E^2 = p^2c^2 + m^2c^4
at low non-relativistic velocities E =p^2/2m + mc^2
(usual kinetic energy + einstein's mass energy)
For a photon always use relativistic formula since they travel w/ velocity of light.
photon mass = 0 (so E62 = p^2c^2)
so yukawa's borrowed energy argument -> can create virtual photons
w/ very low momentum and almost zero total energy.
such virtual photons can travel almost as far as they like
without getting into trouble with the
energy-time uncertainty relation.
thus EM interactions expected to be effective over very large distances.
+: B field don't penetrate far inside a superconductor.
Caused by currents wi/ superconductor screen out/cancel
applied magnetic field in metal (diamagnetic effect)
since no resistance induced currents produce a B field cancels out
all applied magnetic field inside metal.
so B field only penetrates a short distance, as if inside
superconductor photon has very large mass.
to someone living in side sc, might conclue
photons have mass related to distance magnetic fields
can travel in the sc.
+For weak interaction can draw feynman diagram.
In beta decay of neutron a down quark changes into an up quark
emitting a virtual W particle which decays into an electron and an
antineutrino.
but range of the weak force is very small.
So mass of the W particle must be big. The W particle must also
be charged.
Local phase theory of QED massless neutral photon
Theory of weak interactions which need massive charged W's.
+Imagine the vacuum in which we live
is analogous to a "weak superocnductor". Vacuum screening currents can
make it appear that the W particles have mass., Key idea behind
higgs vacuum. In a superconductor, the screening
currents are due to circulating
cooper pairs of electrons. In the case of a guage theory of weak
interactions , these currents are due to Higgs bosons.
+Standard model or GSW model of electroweak interactions predicts that in addition
to the charged W particles, should also exist a neutral heavy Z particle.
a Genuine "weak photon".
predict new quark w/ charm. just as EM force different
for particles w/ different electrical charge, so the strength of
weak force depends on strangeness and charm of the quarks.
+Couldn't do diagrams w/ no closed loops.
loop graphs involve powers of the charge e.
that governs coupling of W and Z's to quarks and leptons.
higher order diagrams , shown by Gerard 't Hooft.
+Havn't found Higgs particle.
Solid-state analogue of higgs particle is a cooper pair of electrons,
thus higgs particle may not be a genuine elementary
particle but a composite.
5d Quarks and gluons
+simple theory of hadronic forces?
quarks in many varieties. Is electroweak force
distinguish between different flavors of quarks? strong force
same whether acts on a strange or a charmed quark.
aside
+Stangeness is well defined physical
quantity. Equivalent quantity called hypercharge.
non-strange have different electric chages.
eigenvalues of the third component of isotopic spin.
use shorthand up and down.
Color = quarks transform according to the
fundamental representation of the special unitary group SU(3).
they carry a color charge.
+Take Omega-
electric charge -1, strangeness -3, spin angular momentum 3/2.
So each quark has spin 1/2, but quarks are fermions
obey pauli exclusion principle, but all quarks in
omega- have same quantum numbers, pauli doesn't allow this.
Each quark has a different color , i.e.
SU(3) threeness means 3 different possible states of the quark.
+QCD = gauge theory based on local phase invariance
of the color properties of quantum amplitudes of the quarks.
EM forces mediated by 0mass gauge particles - photons.
quark-quark interaction described in terms of exchange of strong photons = gluons
photons couple ordinary electric charge of quarks, gluons couple to the
color charge of the quarks. Gluons carry a color charge and the gauge principle
dictates they must interact with themselves.
+so tough to verify QCD since can't isolate a single quark to
witness gluon
5e. Superconductors, magnetic monopoles and quark confinement
+Can't see quarks bythemselves.
+no magnetic monopoles, if so maxwell's equations would
be symmetrical. but not (w/ exchange electric and B fields)
+Dirac showed existence of 1 quantum mechanical magnetic monopole
would imply all electric charges must be exact multiples of the
charge of the electron.
+Two types of superconductor Type-I sc in which B field
is screened out,
type-II sc B field is not entirely expelled from metal, allowed to thread through in thin filaments.
+Also another unexpected result of quantum mechanics
related to bohm-aharanov effect, fact that
magnetic field threading each filament is quantized,
and can only have certain values.
+So quark confinement could come about
vacuum state of QCD is like type-2 superconductor.
QCD has color electric field, color magnetic fields.
A vacuum like a type-2 superconductor allow color
magnetic field to exist in thin tubes.
Quantized amount of B field just right for the field
to begin and end on a color magnetic monopole.
this field line pattern is what we need for
quark confinement.
.............FUNCTIONS
{
<< text expansion , details, mail me for
these if you are interested , thanks! --Ben >>
}
