Transcript The Mystery of Matter: The Course

PHY313 - CEI544
The Mystery of Matter
From Quarks to the Cosmos
Spring 2005
Peter Paul
Office Physics D-143
www.physics.sunysb.edu PHY313
Peter Paul 02/10/05
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What have we learned last time?
mass energy is available to
• Theory of Relativity introduces new • This
produce secondary particles during
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rules that have been proven
experimentally: When v~c…..
A moving clock runs more slowly
as observed by a stationary
observer.
A moving length is shortened in the
direction of flight, as seen by a
stationary observer.
The mass of a body or particle at
rest contains energy E = mc2
This “rest mass” increases with v as
the particle speeds up to the speed
of light.
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decay. (0  2 -rays )
Quantum Mechanics is based on the
fact that particles can act like
waves. They have wavelength and
frequency. They can be trapped like
standing waves on a string.
The square of the wave amplitude
gives the probability of finding the
particle in a certain condition.
When the wave of an electron is
trapped in an atom by the Coulomb
attraction, only discrete standing
waves are allowed.
This leads to quantum states
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Doppler Effect and Red shift
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If a photon is emitted from a moving
source in my direction, do I see any
effect from the moving source?
Yes, if the source is moving toward me,
the source is “pushing” the photon in
my direction. That adds energy to the
photon. Since the energy of the photon
is E = h , the frequency n increases.
If the source, like a star, is moving
away from me the photon loses energy
and n decreases.
This is the famous Red Shift observed
from receding stars and galaxies.
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http://hubblesite.org/newscenter/newsdesk/archiv
e/releases/2004/07
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Wave functions and Tunneling
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http://phys.educ.ksu.edu/vqm/html/qtun
neling.html
• The standing wave inside a
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The sunrays are partially
reflected, but a lot of the
intensity penetrates the coated
lens into the eye.
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potential through is called the wave
function of the particle, . Its
square gives the probability of
finding the particle in a particular
place in space.
If the walls of the trough are not
very high or are very thin, the wave
function can leak out through the
wall.
It has tunneled through the barrier
even though it does not have
enough energy to climb over it.
Electron tunneling is the basis for
most transistors and has huge
practical applications.
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Electrons as Waves on Strings
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1.
2.
3.
Assume electrons are confined in the
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atom over a distance L
Let’s look at it as a linear problem, a
particle as a wave trapped in a box.
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This is like a string of length L that is
fixed at both ends.
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Plucking the string produces standing
waves in the box, with discrete
wavelengths:
 = 2L
 = L = 2L/2
 = 2L/3
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In general  = 2L/n
with n = 1,2,3…

Peter Paul 02/10/05
http://www.cord.edu/dept/physics/p128/lect
ure99_35.html
We call n the principal quantum
number of the system.
Different values of n produce different
energies inside the box:
22
n
p
hc
n
E

2m 2m c2 2L2
2
2 2
As the electron jump form a higher n to
a lower n it looses energy which is
given off as discrete light quanta.
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Electrons in quantized Atoms
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The electrons in an atom are
trapped by the Coulomb potential,
rather than between steep walls.
Thus the energy levels in an atom
are spaced as follows:
.http://physics.ius.edu/~kyle/physlets/quantum
/hydrogen.html
http://www.falstad.com/qmatom/index.html
• (hydrogen wave function)
• How many electrons can I place
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into each level ? This is given by
The Pauli Principle:
No two electrons in an atom can
have the same quantum numbers
Wolfgang Pauli
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
The Spinning Electron
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Then electron spins around its axis
like a spinning top.
• This spin is quantized. Its axis can
only point up or down. Actually, it
wobbles a bit.
S  ms  h
2
 ms 
ms  1/2
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This introduces two more quantum
numbers.
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Filling the atomic levels
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Two electrons can be placed into the
lower level, in accordance with the
Pauli Principle.
• This completes the Helium atom
• Thhe next electrons have to go into
higher levels.
• The electrons can also circle around
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the atom which produces orbital
angular momentum, which is
quantized in turn.
Taking this into account the next
level can take 8 electrons.
This brings the elements up to
Neon.
With this principle we can construct
the entire periodic table!
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The Periodic Table
Th
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The Success of QM and QED
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Quantum Mechanics, the ntheory of
the EM interaction, and its more
general parent, Quantum
Electrodynamics, have become the
most successful theory in science.
Its accuracy is only limited by our
computational capability.
Its applications encompass all of
chemistry, materials science,and
even biology.
New developments in high-end
computation will soon make it
possible to compute at a speed of 1
petaF/s = 1015 executions/second
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The leader in this technology
is IBM in the US and Japan
http://www.research.ibm.com/bl
uegene/
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Ab initio QMD calculations of complex
molecules
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Peter Paul 02/10/05
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
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Protein folding calculated at Stony Brook
From Carlos Simmerling, Chemistry SBU
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Protein Explorer: Petaflops Machine for Molecular
Dynamics simulations (RIKEN-Yokohama)
PC Cluster with with MD-Grape-3 LSI Folding of a small peptide in water from a
random initial conformation by new
chip that calculates the non-bonding simulation algorithm, REMUCA. (Okamoto)
interactions between atoms.
Each MD Grape-3 chip runs at >165
Gflops. 6,144 chips achieve nominal
peak performance of 1 Petaflops.
This goal will be achieved in 2006.
C-Peptide of Ribonuclease A in Water:
In parallel MDM uses Grape chip to
calculate Molecular Dynamics at
speed of 100 Tflops.
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HMCS for Next-Generation HPC simulations:
MPP for Particle Simulation
(GRAPE-6)
MPP for Continuum Simulation Parallel I/O System
PAVEMENT/PIO
(CP-PACS)
…
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32bit PCI
×8
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16 links
16 links
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100base-TX
Switches
Paralel File Server
(SGI Origin2000)
Hybrid System
Communication Cluster
(Compaq Alpha)
(J. Makino)
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Parallel Visualization
Server
Parallel Visualization System
(SGI Onyx2)
PAVEMENT/VIZ
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Magnetic Phenomena in Spherical
Tokamak
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Purpose：
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IRE (Internal Reconnection Event)
observed in ST (Sherical tokamak)
was first simulated by MHD code.
Main results：
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１．Many characteristics observed in
ST (time scale of disruption, strange
configuration change, recovery of
original spherical configuration, etc)
２．Identify the trigger mode (ideal
interchange instability)
３. Mechanism of nonlinear
deformation
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Particle simulation of Tokamak plasma (JT-60)
Origin 3200 512 CPU (0.5 Tflops)
Number of particles： 10 **8
Space resolution ： 160×128×128
Total time steps： 10,000
Elapse time：
one month
Will need several 100 Tflops or 1 Peta Flops computers
for02/10/05
simulating the
ITER plasma
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Heisenberg Uncertainty Relations
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The Heisenberg Uncertainty
Relations follow directly from the
wave character of the electron (or
any other particle). They say…
1. The better I know the position of a
particle the less well I know the
momentum (or mass x velocity) of
the particle. (and vice versa)
x  p 
Werner
Heisenberg
2. The more precisely I know the
time of an event, the less well I
know the energy. (and vice versa)
E t 
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Wave Packet and its Velocity
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A single wave with one frequency
and one wavelength has no
beginning and no end, thus it has no
localization and no specific timing.
• In order to represent a well defined
particle in space, a wave packet has
to be formed by combining many
waves.
• For example if I want to approximate a rectangular wave packet
with length L I need an infinite
number of sine waves.
v
http://www.physics.nwu.edu/ugrad/vpl/wa
ves/index.html
• This bring together a wide range of
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frequencies, which smears out the
momentum.
What is the velocity of such a wave
packet?
Each sine wave has its wave
velocity v(wave) =  x .
The wave packet moves with the
group velocity, which is slower than
v(wave)
v(group) =   x 
This is the velocity of the particle.
L
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Counting wave length and frequency
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The momentum is related to the
wavelength : p = h/ 
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p ~  /2
• To measure  very well I have to
count the distance from wave
maximum to wave maximum over a
very long distance. Thus the
position of the wave in space is
becoming very uncertain.
• Note that h appears in all these
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equations. Thus these are quantum
mechanical effects.
http://www.onr.navy.mil/focus/ocea
n/motion/waves1.htm
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The energy is related to the
frequency:
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E = h   E ~ t
• In order to determine the frequency
with high accuracy, I have to count
wave tops for a very long time.
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Heisenberg in a Box
• Let’s
put an electron into a box and
apply Heisenberg’s principles to it:
• Uncertainty in its location is L.
• Thus from p x L = h
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m v = h/L
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v
• This
means the electron can never sit
still on the bottom as long as it is
confined to the box!
• As L gets smaller v becomes larger.
• The kinetic energy of the particle is
L
• This compares
to 13.6 eV for the
2 2
c
E  2 2 ~ 40eV
2L mc
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For the atom L ~ 0.01 nm
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lowest binding energy of the
electron in the hydrogen atom.
The box size already determines E
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What is the energy scale of the nucleus?
• The nucleus at the center of the
atom has a radius of only ~5 fm. The
distance from the nucleus to the
average location of the lowest
electron is about 1000 radii.
• The nucleus is composed of
neutrons and protons that are held
together by the strong interaction ~
100 times stronger than the
Coulomb force but has a range of
only ~ few fm.
• Thus the n and p can be considered
trapped inside a steep box of 
radius
~10 fm
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• Neutrons and protons are have
masses of about 938 MeV and
behave inside the box otherwise like
electrons.
( c) 2
E  2 2  1MeV
2L mc
• Thus photon transitions in nuclei
are expected to have a million times
higher energy, or frequency.
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Can we extrapolate from atoms to a nucleus?
• The active particles in atoms are the
electrons bound by the long range
Coulomb force to the positive
nucleus.
• The active particles inside a nucleus
must be protons and neutrons which
are bound together by the shortrange nuclear (strong) force.
• Protons and neutrons have masses
around 938 MeV, differing by only
780 keV = 0.08%. Both have spin
1/2 hbar.
• Heisenberg in 1932 conjectured that
protons and neutrons are two
different forms of the same particle,
the nucleon.
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• It was shown by Rutherford that the
nucleus was a dense and hard
medium.Nevertheless…
• It was conjectured already in 1938
and proven in 1952 that nucleons
can be stacked inside a common
potential, like electrons inside the
atom. This is called the shell model.
• How can densely packed nucleons
move around seemingly freely
inside the nucleus. The Pauli
principle provides the answer.
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How are nucleons stacked inside a nucleus ?
• Despite the short range interaction
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the nucleons are stacked inside a
potential well on levels similarly to
the electrons inside atoms.
The Pauli principle allows to place
neutrons and protons separately into
levels. Each have a spin 1/2 hbar. So
4 nucleons fill the lowest level.
This forms the Helium 4 nucleus.
As more and more nucleons go into
the higher levels, the positive charge
of the protons repels new protons
being added. It becomes easier to add
neutrons rather than protons.But
these must go into higher orbitals
which costs energy.
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The combined interaction of all
nucleons is approximated by the
harmonic oscillator well
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Heisenberg and particle decay
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The Mu-meson at rest lives in the
average for 2.3 s. Its mass is 105
MeV.
Mc2() = 206 x mc2(e), thus the 
can decay into an electron.
 e +
The decay is “transacted” by the
weak interaction.
Question for Mr. Heisenberg: How
accurately can we determine the
mass of the  ?
• A neutron is heavier than a proton
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by 782 keV. Thus it can decay into a
proton and an electron.
n  p+ + e- +  with T = 1000 s
This decay has very little extra
energy available:
782 keV - 511 keV = 271 keV
Fortunately the neutrino has almost
zero mass.
6.6 1010 eVs
E  (m c )  
t
2.3s
2
 31010 eV
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How do nuclei and particles decay?
• The Radioactive Decay law says:
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The number of decays/sec is
always proportional to the
number of nuclei/particles that
are available for decay.
Half-life: If there are 1000 nuclei
there that can decay, after a time
T1/2 500 will have decayed. If I
wait again for T1/2, half of the 500,
I.e. 250, will have decayed, and so

on.
Only after an infinite amount of
time will all nuclei have decayed
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dN
   N;
dt
0.63
T1/ 2

• For Pu 240 T1/2 = 6500 yrs
• For every ton available today 1/2
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ton will still be here after 6500
years!
210 = 1024
Thus it will take 10 half-lives to
reduce the Pu from 1 ton to 1 kg
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Third Homework Set, due Feb. 17, 2005
1.
2.
3.
4.
5.
6.
Please draw the fundamental wave and the next harmonic into a box of length
L with infinite high walls. Are standing or running waves? What is the
wavelength of the next harmonic wave?
Why does Mr. Pauli allow us to place only 2 electrons into the lowest atomic
orbit?
How does one produce a wave packet out of waves? Is the speed of the wave
packet different from c? More? Less?
Why can an electron not drop to the bottom of a box and rest there?
Use of an electron beam, rather than a light beam, for microscopy allows to see
much smaller (finer) details. Why is that?
A particle that decays (or a fruit fly that dies) cannot have a sharp energy (or
mass). Why is that? How much does it affect the weight of thy fruit fly?
Please explain.
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How to submit Homework
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2.
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You have 3 possibilities:
Submit it to me in class on the date it is due.
Put it in the TA’s (Xiao Shen) mailbox in the Physics Department
main office on or before the due date
Submit it him by e-mail at the address:
[email protected]
Please DO NOT submit it to me by e-mail
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