1
53717
http://theory.physics.helsinki.fi/~qmii/
Quantum Mechanics II
Spring 2011
Lecturer:
Paul Hoyer
C321
191 50681
Assistants:
Tommi Markkanen (17-31.1, 1.4-6.5)
Samu Kurki
(1.2 - 31.3)
C304
191 50705
Lectures:
Mo 12-14
A315
First lecture on 17 January
We 12-14
A315
TBA
TBA
Exercises:
Problem solutions should be put in the box on the 2nd floor of the A-wing by Friday at 14.00.
Results-->
Exams: 1. TBA
2. TBA
This is an advanced course in quantum mechanics, building on the basic concepts introduced in the course Quantum m
Paul Hoyer Spring 2011
Quantum Mechanics II
Exercise sessions: Time
Paul Hoyer Spring 2011
2
Quantum Mechanics II
Results-->
3
Exams: 1. TBA
Quantum Mechanics II
2. TBA
This is an advanced course in quantum mechanics, building on the basic concepts introduced in the course Quantum mechanics I.
The applications of quantum mechanics to topical issues in modern physics are shown as examples.
The course introduces the general methods required to address these and other topics.
Topics:
1. Basic concepts
2. Rotational symmetry
3. Discrete symmetries
4. Local gauge invariance
5. Second quantization
6. The quantized photon field
7. Relativistic quantum mechanics
8. Quantum field theory
Textbooks:
J. J. Sakurai: Modern Quantum Mechanics (Addison Wesley 1994) [S1]
J. J. Sakurai: Advanced Quantum Mechanics (Addison Wesley 1967) [S2]
I. J. R. Aitchison and A. J. G. Hey: Gauge Theories in Particle Physics, Vol. I:
From Relativistic Quantum Mechanics to QED (IOP Publishing, 3rd Edition, 2003) [AH]
J. Niskanen: Kvanttimekaniikka II (Limes 2003) [N]
Table of Clebsch-Gordan coefficients, spherical harmonics, gradients, Pauli and Dirac matrices.
Quantum Mechanics II in:
Paul
Hoyer Spring
Problems
of 2011
past
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2/2003
1/2004
2/2004
1/2005
2/2005
partial exams: 1/2003
Quantum Mechanics II
4
53717 QUANTUM MECHANICS II / KVANTTIMEKANIIKKA II
10 op
An advanced course in quantum mechanics illustrated with applications.
Textbooks:
Sections 1-4: Sakurai
Modern Quantum Mechanics (1994)
Sections 5-6: Sakurai
Advanced Quantum Mechanics (1967)
Sections 7-8: Aitchison and Hey Gauge Theories in Particle Physics (2003)
1. Central concepts
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
States as vectors in Hilbert space: bra-ket notation
Operators: hermitian, eigenvalues and eigenstates
Completeness and orthonormality of eigenstates
Unitary transformations: cf. Fourier transform
Time development: cf. electron spin precession
Non-crossing eigenvalues: MSW effect for solar neutrinos
Heisenberg picture: eqs of motion, Ehrenfest’s theorem
The interaction picture S-matrix
Propagator cf. free particle and harmonic oscillator
Path integral cf. neutron in gravitational field of the Earth
2. Rotational symmetry
Basics
Paul Hoyer1.Spring
2011 of
group theory: SO(3)
Quantum Mechanics II
5
Measurement of the muon magnetic moment
Brookhaven National Lab., E821
http://www.g-2.bnl.gov/
Paul Hoyer Spring 2011
Quantum Mechanics II
6
gµ/2 = 1.0 011 659 214 (8)(3) e /2mµ
α
=
2π
0.0011614
Paul Hoyer Spring 2011
Quantum Mechanics II
magnet normally behaves. We tested the behaviour using muons implanted into silver in
7
sample (rather than into the sample itself). If the sample behaved like a permanent
magnet
http://www.isis.stfc.ac.uk/
would penetrate the silver and be detectable by the muons. This is indeed what is revealed –
Sn O MuSR
does indeed
behave like a permanent magnet below its transition temperature of
science
Muon Spin Rotation
2 7
The MuSR spectrometer can be used for a wide variety of muon experiments. It is mainly used to study
magnetic and superconducting materials, but is also used to study charge transport, proton diffusion,
and the chemistry of muon radicals.
Some examples of recent research carried out using MuSR are shown below:
An outsider’s view: a novel muon study of frustration
SR Giblin, JDM Champion (ISIS), HD Zhou and CR Wiebe (Florida State University, USA), JS
Gardener (NIST, USA), I Terry (Durham University), S Calder, T Fennell, ST Bramwell (University
College London)
Contact: Dr Sean Giblin
Further reading: SR Giblin et al., Phys. Rev. Lett. 101 (2008) 237201
Frustration occurs when it is not possible to satisfy all interactions. For example, a magnetic atom
might want its spin direction to be misaligned with that of a neighbouring atom (if the interactions are
antiferromagnetic). But, for some arrangements of atoms, it’s possible to find that misalignment with
one neighbour prevents misalignment with another – producing frustration. Frustration plays an
important role in a diverse range of physics, from magnetism to protein folding. Pyrochlores – magnetic
materials with
atoms arranged
in a particular way that leads to frustration – are fascinating as by
Rutherford
Appleton
Laboratory
About ISIS
Science
Instruments
Groups
People
User office
Apply for beamtime
Beam Status
Learning
changing
one
atom
the
frustration
behaviour
changes,
culminating
in
properties
such
as a ‘spin liquid’,
near Oxford in the United Kingdom
‘spin glass’ or ‘spin ice’.
Home
Instruments
Musr
Science
has
previously
led some
tomagnetic
believe itfields
exhibits a novel state of
frustration
Tb Sn
An The
oscillatory
signalininpyrochlore
the muon data
is a O
clear
indication
of static
internal
2 2 7
Science
in Tb2Sn2O7. The inset shows the temperature dependence of the internal field below the
magnetism
in which the magnetisation direction reverses multiple times a second. This is not how a
Documents
Paul Hoyer
Spring 2011
Quantum Mechanics II
transition.
Nuclear magnetic resonance imaging: MRI (aka NMR)
8
MRI image of knee
Commission prepares to revise MRI directive
Magnetic resonance imaging researchers have welcomed an investigation into the European Commission’s
Physical Agents directive, which they claim would substantially limit the procedures that can be carried out.
A multinational consortium, led by the Finnish Institute of Occupational Health, was chosen by the
Commission on 23 December 2008 to carry out an impact assessment of the legislation. The consortium,
known as FICETTI, will consider five options ranging from not changing the directive to scrapping it
entirely.
Paul Hoyer Spring 2011
Quantum Mechanics II
Press releases of the Alliance for MRI
December 20, 2010
http://www.myesr.org/html/img/pool/ 9
20122010_Alliance_Press_statement_
delay_of_draft_EMF_Directive_EN.pdf
Alliance for MRI says delay in new rules
creates uncertainty for patients
Alliance for MRI says delay in new rules creates uncertainty for patients
Brussels, Dec. 20, 2010 - Patients face continued uncertainty because of delay in a
proposed revision of European Union rules governing the use of MRI scanners, a key
medical tool for detecting disease, an official of the Alliance for MRI said on Monday.
A delay in completing an impact assessment of the possible changes has forced the
European Commission to wait until next year instead of completing work by month’s
end, the Alliance has been told by those involved in the process.
“Magnetic Resonance Imaging is a vital tool for the diagnosis and treatment of brain
diseases, and for related research. We hope the European Commission, the
European Parliament, and Member States will redouble their efforts for a quick
solution to this problem,” said Mary Baker, president of the European Federation of
Neurological Associations, a founding member of the Alliance for MRI.
Paul Hoyer Spring 2011
Quantum Mechanics II
http://arxiv.org/abs/0901.3443
10
Solar neutrino detection
Authors: Lino Miramonti
(Submitted on 22 Jan 2009)
Abstract: More than 40 years ago, neutrinos where conceived as a way to test
the validity of the solar models which tell us that stars are powered by nuclear
fusion reactions. The first measurement of the neutrino flux, in 1968 in the
Homestake mine in South Dakota, detected only one third of the expected value,
originating what has been known as the Solar Neutrino Problem. Different
experiments were built in order to understand the origin of this discrepancy. Now
we know that neutrinos undergo oscillation phenomenon changing their nature
traveling from the core of the Sun to our detectors. In the work the 40 year long
saga of the neutrino detection is presented; from the first proposals to test the
solar models to last real time measurements of the low energy part of the neutrino
spectrum.
Comments: 8 pages, 5 figures. III School on Cosmic Rays and Astrophysics August 25 to
September 5, 2008 Arequipa (Peru) AIP conference proceeding
Paul Hoyer Spring 2011
Quantum Mechanics II
11
Paul Hoyer Spring 2011
Quantum Mechanics II
12
http://arxiv.org/abs/0901.2505
!e‘s convert to !µ or !! with confidence level (CL) of more than 7" .
_
– KamLAND find that reactor !e disappear over distances of about 180 km and they observe a
– Solar
distortion of their energy spectrum. Altogether their evidence has more than 3" CL.
– The evidence of atmospheric (ATM) !µ disappearing is now at > 15", most likely
converting to
!! .
– K2K observe the disappearance of accelerator !µ’s at distance of 250 km and find a
distortion of their energy spectrum with a CL of 2.5–4".
– MINOS observes the disappearance of accelerator !µ’s at distance of 735 km and find
a distortion of their energy spectrum with a CL of # 5".
Paul Hoyer Spring 2011
Quantum Mechanics II
13
http://arxiv.org/abs/0901.2505
_
Allowed regions for 2! oscillations of solar !e (left) and KamLand !e (right).
The different contours correspond to the allowed regions at 90%, 99% and 3"
CL.
Paul Hoyer Spring 2011
Quantum Mechanics II
14
http://arxiv.org/abs/0901.2505
Allowed regions from the analysis of atmospheric !µ data (left), K2K (central)
and Minos (right). The different contours correspond to the allowed regions at
90%, 99% and 3" CL.
Paul Hoyer Spring 2011
Quantum Mechanics II
!e mass limits from decay measurements
15
http://cupp.oulu.fi/neutrino/nd-mass.html
The KATRIN experiment is designed to measure
the mass of the electron neutrino directly with a
sensitivity of 0.2 eV. It is a next generation tritium
beta-decay experiment scaling up the size and
precision of previous experiments by an order of
magnitude as well as the intensity of the tritium
beta source.
http://www-ik.fzk.de/~katrin/index.html
Paul Hoyer Spring 2011
Quantum Mechanics II
16
Physics Today November 2000
Recommended reading for QM II!
Paul Hoyer Spring 2011
http://theory.physics.helsinki.fi/~qmii/QCrypt_PT.pdf
Recent article on quantum information in Arkhimedes:
http://theory.physics.helsinki.fi/~qmii/Arkhimedes_08-1_mottonen.pdf
Quantum Mechanics II
17
Home
Products
Customers
TM
Research Labs
MAGIQ QPN 8505
Careers
About Us
Security Gateway
TM
Uncompromising VPN Security
http://www.magiqtech.com/MagiQ/Products.html
Products
Carrier Class Network Security
• Quantum Key Distribution
MagiQ offers the following family of products on a custom specification and order basis. Please contact
OVERVIEW
661-8300
x201 (andy@magiqtech.com) for more information.
• Key refresh rate up to 100 256 bit keys
Andrew
a second Hammond at (617)
• Intrusion Detection
Technology companies have responded to heightened requirements for information security with an avalanche
• Always-on base VPN
of new cryptography products. As sophisticated as these solutions are, they have this in common: they all rely
• Data protection with best of breed
encryption technologies
on computational difficulty as the source of their protection and none can escape the fact that information security that
• In-transit data security
relies on computational difficulty is ultimately vulnerable.
Fiber Optic Sensors (MFOS - 415OL)
• Network compatibility
Breaking through to information security that’s NOT VULNERABLE means breaking new ground. Breaking with the
• Implements industry standards
BB84, 3DES,
of refining current
solutions; finding
new directions;
pioneering completely
thatmost
can’t be
MagiQ
hasAES
unique capabilities for tradition
manufacturing
optical-vector
matched
interferometers
for usenew
in technologies
the world’s
• demanding
Integrated encryption
andapplications.
Quantum
threatened
by the next
new chip
or the nextlinearity,
new software
algorithm:
that’srange,
been the broadband
mission of MagiQ
Technologies
sensor
This
process
enables
extreme
high
dynamic
frequency
Key Distribution
one enclosure
since
response,
andinhigh
sensitivity from
the1999.
sensor. These sensors are EMI free/immune and completely passive making
compatible
with
high temperature or harsh environments. Optical-vector matching allows multiple sensors to be
• them
Quantum
Key Distribution
compatible
with DWDMfor phased array applications.
combined
Presenting MagiQ QPN™ Security Gateway Quantum Key Distribution (QKD) System.
• Low
total Spring
cost of ownership
Paul
Hoyer
2011
Quantum Mechanics II
18
From Sakurai: Modern Quantum Mechanics
Paul Hoyer Spring 2011
Quantum Mechanics II
ultracold neutrons. The reactor neutrons are cooled by being made to traverse a liquid-deuterium moderator
and then work their way uphill
against gravity through piping whose
reflecting walls preferentially absorb
out the more energetic neutrons. The
resulting ultracold neutrons, collimated and concentrated in momentum space by turbines, reach
Nesvizhevsky’s gravitational well as a
horizontal beam with a sprinter’s
speed of about 10 m/s. But in the vertical direction, transverse to the
beam, the effective temperature of the
neutron aggregation is only 20 nK,
corresponding to an energy of a peV or
so.
The experiment’s one-dimensional
gravitational potential well is shown
in figure 1, together with the calculated wavefunctions (squared) of its
three lowest-lying neutron energy
eigenstates. The one dimension is the
height z above the horizontal slab of
material that serves as a perfect
reflecting mirror for the ultracold
neutrons and thus as an impenetrable
wall of the potential well. At sufficiently low temperatures, many metal
surfaces are perfect neutron reflectors
at all angles of incidence.
The second, sloped wall is the
potential energy
V(z) = mgz
due to the gravitational field itself,
where m is the neutron mass and g is
22 MARCH 2002 PHYSICS TODAY
Paul Hoyer Spring 2011
the horizontal beam of
cold neutrons at the
mouth of a 10-cm-long gap between the
mirror and an upper surface that
either absorbs or scatters out of play
any neutron that touches it.2 (See figure 2.) The vertical spacing between
the mirror and the absorber–scatterer
can be varied from 0 to 100 mm. A neutron detector at the far end of the gap
measures the neutron flux that gets
through as a function of the gap height.
The beam is directed at the gap with
a slight upward tilt. The range of neutron energies emerging from the ultracold source leaves a significant spread
in the vertical-velocity distribution of
the entering beam. The gap height
then serves as a filter that sets an
zontal beam velocity. So, if the exper19
imenters have taken adequate care to
eliminate any mechanical or magnetic effects that might couple the velocity components, one expects that the
transmission of neutrons through the
gap as a function of its height will
exhibit a threshold at about the width
of the ground-state wavefunction, followed by a sequence of steps at
heights corresponding to the widths of
the excited-state wavefunctions.
The red curve in figure 3 is the
detailed prediction of the neutron flux
reaching the detector. The blue curve,
with no threshold, is what one would
expect simply from geometric and
phase-space considerations, without
Absorber–scatterer
Detector
Neutron mirror
Collimator
10 cm
FIGURE 2. SCHEMATIC SETUP of the experiment that detected quantum states of
neutrons in a gravitational potential well. A horizontal beam of ultracold neutrons
with, on average, a slight upward tilt enters a long, narrow gap of adjustable height
between a neutron mirror floor and an absorbing–scattering ceiling. The gap height,
much exaggerated in this drawing, limits the vertical velocity component of the
entering neutrons. The detector records the fraction of transmitted neutrons as a
Physics
Today,
March
2002, page 20
function of the gap height.
(Adapted
from ref.
1.)
~qmii/QM+Gravity_PT55(02).pdf
http://www.physicstoday.org
Quantum Mechanics II
http://arxiv.org/abs/hep-ph/0602093
A quantum mechanical description
of the experiment on the observation of gravitationally bound states
20
Quantum states in the Earth’s gravitational field were observed, when ultra-cold neutrons fall under gravity. The experimental
results can be described by the quantum mechanical scattering model as it is presented here. We also discuss other geometries of the
experimental setup which correspond to the absence or the reversion of gravity. Since our quantum mechanical model describes,
particularly, the experimentally realized situation of reversed gravity quantitatively, we can practically rule out alternative
explanations of the quantum states in terms of pure confinement effects.
Schematic view with mirrors, absorber and quantum
mechanical boundary conditions. In the experiment, one
mirror of length 10 cm or, as an option as shown here, two
bottom mirrors of length 6 cm were used.
Paul Hoyer Spring 2011
Circles: Data from the 2nd run 2002 with one bottom
mirror [16]. Solid: Transmission coefficient from the
phenomenological scattering model. Dash: The
classical expectation for the neutron transmission
coefficient.
Quantum Mechanics II
21
http://capp.iit.edu/hep/pbar/Phillips-GravityExpt.pdf
Paul Hoyer Spring 2011
Quantum Mechanics II
22
http://capp.iit.edu/hep/pbar/Phillips-GravityExpt.pdf
Paul Hoyer Spring 2011
Quantum Mechanics II
MEASURINGHOME
g with
a beam of
antihydrogen
COLLABORATION
EXPERIMENT
PUBLICATIONS
23
CONTACTS
INTERNAL
http://aegis.web.cern.ch/aegis/home.html
WELCOME TO AEgIS
Does antimatter
fall down?
AEgIS is a physics experiment that takes place at the european laboratory CERN, using the
by OF
the AD
accelerator. AEgIS is a collaboration of physicists
from allfall
Does antimatter
MEASURINGantiprotons
g WITH delivered
A BEAM
ANTIHYDROGEN
around the world.
down?
AEgIS is a physics experiment that takes place at the european laboratory
The primary scientific goal of the AEgIS experiment is the direct! measurement of the Earth’s
CERN, using the
antiprotons
delivered
byantihydrogen.
the AD accelerator.
AEgISof istheaexperiment, a gravity
gravitational
acceleration
g on
In the first phase
collaboration ofmeasurement
physicists from
all around
with 1%
precisionthe
willworld.
be carried out by sending an antihydrogen beam through a
classical Moire deflectometer coupled to a position sensitive detector. This will represent the first
direct measurement of a gravitational effect on an antimatter system.
PaulThe
Hoyerprimary
Spring 2011scientific goal of the AEgIS experiment is the direct
Quantum Mechanics II
24
CLEBSCH-GORDAN
COEFFICIENTS
1
0
+1
1
0
0
+ 1/2 + 1/2 1
+ 1/2 ! 1/2 1/2 1/2 1
! 1/2 + 1/2 1/2 ! 1/2 ! 1
1/2 " 1/2
N ota ti on:
! 1/2 ! 1/2 1
m2
m1
m2
.
.
.
3/2
+ 3/2 3/2 1/2
1 + 1/2 + 1/2
+ 1 + 1/2
1 " 1/2
+ 1 ! 1/2
0 + 1/2
m1
.
.
.
M
M
...
...
Note: A square root should be taken of
each element shown (apart from sign).
2/3 1/3 3/2
1/3 ! 2/3 ! 3/2
! 1 ! 1/2
J
C oefficients
1/3 2/3 3/2 1/2
2/3 ! 1/3 ! 1/2 ! 1/2
0 ! 1/2
! 1 + 1/2
J
1
More complete table of C-G
coefficients, spherical functions etc.
may be found on the QM II home page
!j1 j2 m1 m2 |j1 j2 JM "
Paul Hoyer Spring 2011
= (−1)J−j1 −j2 !j2 j1 m2 m1 |j2 j1 JM "
Quantum Mechanics II
25
Paul Hoyer Spring 2011
Quantum Mechanics II
26
8 .0 0
Spin
7 .0 0
P.Desgrolard, M.Giffon, E.Martynov, E.Predazzi, hep-ph/0006244
Regge trajectory
f (2 5 1 0 )
6
2
R e ! (m )
6 .0 0
a 6 (2 4 5 0 )
" 5 (23 5 0)
5 .0 0
f 4 (2 05 0)
a (2 02 0)
4 .0 0
4
" 3(1 70 0 )
# (1 67 0 )
3 .0 0
3
2 .0 0
1 .0 0
0 .0 0
0 .0 0
f 2(1 2 7 0 )
a 2 (1 3 1 8 )
" (77 0 )
#(7 82 )
2 .0 0
4 .0 0
m
2
For unknown
reasons, spins of
elementary
particles are
proportional to
their mass2
6 .0 0
8 .0 0
1 0 .0 0
2
(G eV )
Figure 1: Chew-Frautschi plot for the fully exchange-degen erate f , ω, ρ and a 2 trajectories.
The solid line denotes the trajectory with the parameters obtained in our fit; the dashed line
is the trajectory α(m2) = 0.48 + 0.88m2 (m in GeV).
Paul Hoyer Spring 2011
Quantum Mechanics II
27
This historical review of the discovery of
parity violation may be found at:
http://ccreweb.org/documents/parity/parity.html
Paul Hoyer Spring 2011
Quantum Mechanics II
28
The positive pion, a spinless particle, initially has zero angular momentum and zero linear momentum (we consider only pion decay at rest). Therefore linear
momentum conservation requires that the decay products (a positive muon and a muon neutrino) are emitted in opposite directions with equal and opposite
momenta; meanwhile, angular momentum conservation requires that they have equal and opposite spin. The weak interaction governing this decay process has the
remarkable property that it creates only ``left-handed'' or ``negative helicity'' neutrinos (i.e. having their spin and angular momentum in opposite directions, as
shown) and ``right-handed'' or ``positive helicity'' antineutrinos, so the !+ must also have its spin pointing back along its momentum. This gives a beam of perfectly
spin-polarized muons from pion decay - an essential ingredient for !SR.
Γ(π → eνe )
= 1.23 · 10−4
Γ(π → µνµ )
The mirror-image reaction never occurs in nature, because the parity inversion performed by the mirror changes a left-handed neutrino to a right-handed one, which
the weak interaction cannot produce.
Paul Hoyer Spring 2011
http://musr.org/~jess/musr/cap/pidk.htm
Quantum Mechanics II
http://www.mpi-hd.mpg.de/hfm/CosmicRay/Showers.html
Paul Hoyer Spring 2011
29
Quantum Mechanics II
Antoine Weis:
Atomic physics tests of the Standard Model
http://theory.physics.helsinki.fi/~qmii/Lecture1.pdf
http://theory.physics.helsinki.fi/~qmii/Lecture2.pdf
30
THE STANDARD MODEL
· charged currents and neutral currents
THEORY OF PARITY VIOLATION IN ATOMS
· parity violating asymmetry
· parity violating potential and matrix elements
PARITY VIOLATION IN ATOMS: GENERAL CONSIDERATIONS
· allowed and forbidden transitions
· parity violating electric dipole amplitude
· classical representation of PV atom
· q. m. representation of PV atom
· M1 - E1pv interference
· optical rotation experiments
SPECIFIC PARITY VIOLATION EXPERIMENTS IN ATOMS
· optical rotation experiments
· experiments with 133Cs
- field-free circular dichroism
Paul Hoyer Spring 2011
Quantum Mechanics II
31
http://theory.physics.helsinki.fi/~qmii/bouchiat.pdf
Paul Hoyer Spring 2011
Quantum Mechanics II
http://en.wikipedia.org/wiki/Muon-catalyzed_fusion
http://www.triumf.ca/welcome/h-fusion.html
32
Muon Catalysed Fusion
p
Proton
n
Neutron
p n
+
Compound
molecule
Paul Hoyer Spring 2011
n n
p
p
n
+
Muonic
tritium nucleus
Tritium
nucleus
Free
muon
n n
p
n n
p
n n
p p
Alpha
particle
+
n
Free
neutron
Deuterium
nucleus
+
+
Energy
from
fusion
Free
muon
Quantum Mechanics II
33
http://en.wikipedia.org/wiki/Electronic_band_structure
Comparison of the electronic band structures
of metals, semiconductors and insulators
Paul Hoyer Spring 2011
Quantum Mechanics II
http://en.wikipedia.org/wiki/Neutron_electric_dipole_moment34
Transformations of magnetic and electric
dipole moments under parity and time reversal
E P
B !
-E
B
E T
B !
E
-B
A permanent electric dipole moment of a
fundamental particle violates both parity (P) and
time reversal symmetry (T). This is quickly
comprehensible by looking at the neutron with its
magnetic dipole moment and hypothetical electric
dipole moment. Under time reversal, the magnetic
dipole moment changes its direction, whereas the
electric dipole moment stays unchanged. Under
parity, the electric dipole moment changes its
direction but not the magnetic dipole moment. As
the resulting system under P and T is not
symmetric with respect to the initial system, these
symmetries are violated in the case of the
existence of an EDM. Having also CPT
symmetry, the combined symmetry CP is violated
as well.
Paul Hoyer Spring 2011
Quantum Mechanics II
http://en.wikipedia.org/wiki/Neutron_electric_dipole_moment35
Paul Hoyer Spring 2011
Quantum Mechanics II
http://www.hitachi.com/rd/research/em/abe.html
36
Paul Hoyer Spring 2011
Quantum Mechanics II
Verification of the Aharonov-Bohm effect
Paul Hoyer Spring 2011
37
http://www.hitachi.com/rd/research/em/abe.html
Quantum Mechanics II
38
Paul Hoyer Spring 2011
Quantum Mechanics II
Helsinki University of Technology
SQUID Magnetometers
39
122-SQUID neuromagnetometer
In the middle of 1992, a sophisticated 122channel brain research system became
operational. This instrument is the
culmination of more than ten years of
development of the magnetoencephalographic (MEG) technology in the LTL. The
magnetic field caused by neural currents
flowing in the brain is measured by an array
of 122 superconducting sensors which cover
the subject's head in a helmetlike fashion.
http://ltl.tkk.fi/triennial/squid.html
Paul Hoyer Spring 2011 http://hyperphysics.phy-astr.gsu.edu/HBASE/Solids/Squid.html
Quantum Mechanics II
• A Personal Witness Account of the Keppe
40
http://arxiv.org/abs/hep-ex/0302011
Motor
• Enviro Energies' magnetically levitated
Mag-Wind
www.magnetmonster.de
"Free Energy"
You are here: PureEnergySystems.com > News > June 5, 2004
Googling, you also find this:
ws
aily FE News
atures
ESN Specials
ee Energy Now
is Week in FE
ewsletter
bmit
PureEnergySystems.com > News > June 5, 2004
High Energy Magnetic Monopole
Sequestered by U.S. Government
S
With resistive forces of at least 10 to 20 tons per square meter at 1/4" thickness, this
material could power a device approximately the size of standard can of oil, delivering
a minimum torque ratio of 1.5 tons of turning force. Witness comes forward after
thirty years despite threats on his life.
Paul Hoyer Spring 2011 by James D. Fauble
Plugs in
and o
Quantum Mechanics II
Harmonic Oscillator
wave functions
Paul Hoyer Spring 2011
41
Quantum Mechanics II
http://www.hep.man.ac.uk/babarph/babarphysics/positron.html
42
This is a picture of one of the first positron tracks observed by
Anderson in 1933. It was taken in a cloud chamber in the
presence of a magnetic field (so the particle paths are curved).
A cloud chamber contains a gas supersaturated with water
vapour. In the presence of a charged particle (such as a
positron), the water vapour condenses into droplets - these
droplets mark out the path of the particle.
The band across the middle is a lead plate, which slows down
the particles. The radius of curvature of the track above the plate
is smaller than that below. This means that the particle is
travelling more slowly above the plate than below it, and hence
it must be travelling upwards. From the direction in which the
path curves one can deduce that the particle is positively
charged. That it is a positron and not a proton can be deduced
from the long range of the upper track - a proton would have
come to rest in a much shorter distance.
Carl Anderson won the 1936 Nobel Prize for Physics for this
discovery.
Paul Hoyer Spring 2011
Picture taken from C.D. Anderson, Physical Review 43, 491
(1933).
Quantum Mechanics II
http://teachers.web.cern.ch/teachers/archiv/HST2005/bubble_chambers/BCwebsite/index.htm 43
Bubble Chamber
Paul Hoyer Spring 2011
Quantum Mechanics II
Electrons, positrons and photons
44
The knock-on electron (bottom left) and
the lone Compton electrons show that
negative particles turn to the left.
There are three linked highlighted examples of high energy photons materialising into e+e– pairs in the field
of a nucleus. In the order in which it have happened:
* the first photon materialises (nearest the bottom of the picture);
* the second is most likely a bremsstrahlung photon from the of the first e+e– pair;
* the Compton electron (on the right of the picture) is caused by a bremsstrahlung photon from the e+of the second pair;
* the third e+e– pair (on the left of the picture) is caused by a bremsstrahlung photon from the of the second e+e– pair.
The thick track coming in from the top of the picture (one can tell which way it is going by noticing the
knock-on electron) is a cosmic ray, probably a muon. This is a reminder of the link between cosmology and
particle physics.
Paul Hoyer Spring 2011
Quantum Mechanics II
http://teachers.web.cern.ch/teachers/archiv/HST2005/bubble_chambers/BCwebsite/index.htm 45
A classic example of a pi mu e decay
!+
+
e+
µ+
This picture was taken in the CERN 2m hydrogen bubble chamber. (We think the incoming beam consists of
K+ particles at 10 GeV/c.)
The little curly electron near the collision point tells us that negative particles turn to the left.
The track that starts going to the right before looping round is a !+. It stops and decays to a µ+ and a muon
neutrino "µ. The muon can only receive about 30 MeV/c (for details click here) in this decay and can only
travel about 1 cm in hydrogen before it , itself, stops. It then decays into a positron (which spirals
characteristically), an electron neutrino "e and a muon-antineutrino
.
Paul Hoyer Spring 2011
Quantum Mechanics II
46
The QED experience
Paul Hoyer Spring 2011
Quantum Mechanics II
Physical Review 140 (1965) B397
47
In his report to the 12th Solvay Congress (Brussels, 1961) on “The Present Status of Quantum
Electrodynamics” (QED), Feynman called for more insight and physical intuition in QED
calculations. To quote from a particularly relevant passage:
“It seems that very little physical intuition has yet been developed in this subject. In nearly
every case we are reduced to computing exactly a coefficient of some specific term. We have
no way to get a general idea of the result to be expected. To make my view clearer, consider,
for example, the anomalous electron moment, (g–2)/2 = #/2! – 0.328#2/!2 . We have no
physical picture by which we can easily see that the correction is roughly #/2! , in fact, we do
not even know why the sign is positive (other than by computing it). In another field we
would not be content with the calculation of the second order term to three significant figures
without enough understanding to get a rational estimate of the order of magnitude of the third.
We have been computing terms like a blind man exploring a new room, but soon we must
develop some concept of this room as a whole, and to have some idea of what is contained in
it. As a specific challenge, is there any method of computing the anomalous moment of the
electron which, on first rough approximation, gives a fair approximation to the # term and a
crude one to #2 ; and when improved, increases the accuracy of the #2 term, yielding a rough
estimate of #3 and beyond?”
Paul Hoyer Spring 2011
Quantum Mechanics II
48
Paul Hoyer Spring 2011
Quantum Mechanics II
Bound states of atoms
Paul Hoyer Spring 2011
U.D. Jentschura et al,
PRL 95 (2005) 163003
49
Quantum Mechanics II
The accuracy of measurement and theory
50
Many of our most accurate predictions come from QED atoms.
For example, the 2S1/2 – 8S1/2 splitting in Hydrogen:
$(2S1/2 – 8S1/2)H = 770 649 350 012.0(8.6) kHz EXP
= 770 649 350 016.1(2.8) kHz QED
U.D. Jentschura et al,
PRL 95 (2005) 163003
The QED result is based on perturbation theory:
– an expansion in # = e2/4! % 1/137.035 999 11(46)
However, the series must diverge since for any # = e2/4! < 0 the electron
charge e is imaginary: The Hamiltonian is not hermitian and probability not
conserved.
F. Dyson
The perturbative expansion is believed to be an asymptotic series.
The good agreement with QED seems fortuituous, from a purely
theoretical point of view.
For a recent discussion of the truncation effects
in asymptotic expansions see Y. Meurice, hep-th/0608097
Paul Hoyer Spring 2011
Quantum Mechanics II
51
Feynman rules
Paul Hoyer Spring 2011
Quantum Mechanics II
52
Paul Hoyer Spring 2011
Quantum Mechanics II
http://arxiv.org/abs/0809.2846
53
Abstract
Paul Hoyer Spring 2011
Quantum Mechanics II
REVIEW OF MODERN PHYSICS, VOLUME 83, JANUARY–MARCH 2011
54
Data tables for Lorentz and CPT violation
V. Alan Kostelecký
Physics Department, Indiana University, Bloomington, Indiana 47405, USA
Neil Russell
Physics Department, Northern Michigan University, Marquette, Michigan 49855, USA
(Received 6 March 2010; published 10 March 2011)
This work tabulates measured and derived values of coefficients for Lorentz and CPT violation in
the standard-model extension. Summary tables are extracted listing maximal attained sensitivities
in the matter, photon, and gravity sectors. Tables presenting definitions and properties are also
compiled.
V. Alan Kostelecký and Neil Russell: Data tables for Lorentz and CPT violation
12
DOI: 10.1103/RevModPhys.83.11
TABLE I.
Type
CONTENTS
PACS numbers: 11.30.Cp, 11.30.Er
List of tables.
Table
Content
mass dimension 4 or less. The corresponding coefficients
Lorentz violation are dimensionless or have positive m
I. Introduction Data
11
V
Electron sector
dimension.
VI
Proton sector
II. Summary Tables
12
VII
Neutron sector
The results summarized here concern primarily but
VIII
Photon sector
III. Data Tables
15
IX
Charged-lepton sector
exclusively the coefficients for Lorentz violation in the m
IV. Properties Tables
20
X
Neutrino sector
XI
Meson sector
mal SME. We compile data tables for these SME coefficie
A. Minimal QED extension XII
22
Electroweak sector
XIII
Gluon sector
including both existing experimental measurements and so
B. Minimal SME
28
XIV
Gravity sector
theory-derived limits. Each of these data tables prov
XV
Nonminimal photon sector
C. Nonminimal photon sector
29
Properties
XVI
Lagrange density for the minimal QED extension
in Riemannabout
spacetimethe results of searches for Lorentz viola
information
XVII
C, P, T, properties for operators for Lorentz violation in QED
XVIII
Definitions for the fermion sector of the for
minimal
extension sector of the SME. For each measuremen
a QED
specific
XIX
Definitions for the photon sector of the minimal QED extension
constraint,
list the spacetime
relevant coefficient or combina
XX
Lagrange density for the fermion sector of
the minimal SMEwe
in Riemann-Cartan
I. INTRODUCTION
XXI
Lagrange density for the boson sector of the minimal SME in Riemann-Cartan spacetime
of coefficients, the result as presented in the literature,
XXII
Coefficients in the neutrino sector
XXIII
Quadratic Lagrange density for the nonminimal photon sector in Minkowski spacetime
which spacetime
the search was
performed,
and II
the sou
XXIV
Spherical
for the nonminimalcontext
photon sectorin
in Minkowski
Paul Hoyer Spring
Quantum
Mechanics
Recent2011
years have seen a renewed
interest
in coefficients
experimental
Summary
II
III
IV
Maximal sensitivities for the matter sector
Maximal sensitivities for the photon sector
Maximal sensitivities for the gravity sector
http://www.lnf.infn.it/~levisand/graal/graal_beam_intro.html
55
Backscattered photon beam at ESF (Grenoble):
6 GeV electrons against laser photons
Paul Hoyer Spring 2011
Quantum Mechanics II
http://www.cambridge.org/resources/052182379X/2064_ch06.pdf
Paul Hoyer Spring 2011
56
Quantum Mechanics II
Lattice waves
http://en.wikipedia.org/wiki/Phonon
57
Diagonalized
Collection of N+1 independent
harmonic oscillators, labelled by k
Paul Hoyer Spring 2011
For each k, state is labelled by number nk of phonons
Quantum Mechanics II
http://www.cambridge.org/resources/052182379X/2064_ch06.pdf
Paul Hoyer Spring 2011
58
Quantum Mechanics II
http://www.cambridge.org/resources/052182379X/2064_ch06.pdf
59
Phonons, photons, pions can be created and annihilated in scattering
_
e+e– & q q g :
Paul Hoyer Spring 2011
'*
Feynman
diagram
Quantum Mechanics II