Peter Paul 02/10/05PHY313-CEI544 Spring-051 PHY313 - CEI544 The Mystery of Matter From Quarks to the...

Peter Paul 02/10/05 PHY313-CEI544 Spring-05 1

PHY313 - CEI544The Mystery of Matter

From Quarks to the CosmosSpring 2005

Peter Paul

Office Physics D-143

www.physics.sunysb.edu PHY313


What have we learned last time?

• Theory of Relativity introduces new 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.

• This mass energy is available to produce secondary particles during 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


Doppler Effect and Red shift

• 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.

• http://hubblesite.org/newscenter/newsdesk/archive/releases/2004/07


Wave functions and Tunneling

• http://phys.educ.ksu.edu/vqm/html/qtunneling.html

• The standing wave inside a 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.

The sunrays are partially reflected, but a lot of the intensity penetrates the coated lens into the eye.


Electrons as Waves on Strings

• Assume electrons are confined in the atom over a distance L

• Let’s look at it as a linear problem, a particle as a wave trapped in a box.

• This is like a string of length L that is fixed at both ends.

• Plucking the string produces standing waves in the box, with discrete wavelengths:

1. = 2L

2. = L = 2L/2

3. = 2L/3

In general = 2L/n

with n = 1,2,3…

• http://www.cord.edu/dept/physics/p128/lecture99_35.html

• We call n the principal quantum number of the system.

• Different values of n produce different energies inside the box:

• As the electron jump form a higher n to a lower n it looses energy which is given off as discrete light quanta.

€

E =p2

2m=

h2c 2

2mc 2

n2

2L( )2

n2


Electrons in quantized Atoms

• 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 into each level ? This is given by

• The Pauli Principle:

No two electrons in an atom can have the same quantum numbers

Wolfgang Pauli


The Spinning Electron

• 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.

• This introduces two more quantum numbers.

€

S = ms • h2π = ms • h

ms = ±1/2


Filling the atomic levels

• 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 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!


The Periodic Table

Th


The Success of QM and QED

• 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

The leader in this technology is IBM in the US and Japan

http://www.research.ibm.com/bluegene/


Ab initio QMD calculations of complex molecules

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.


Protein folding calculated at Stony Brook

From Carlos Simmerling, Chemistry SBU


Folding of a small peptide in water from a random initial conformation by new simulation algorithm, REMUCA. (Okamoto)

C-Peptide of Ribonuclease A in Water:

Protein Explorer: Petaflops Machine for Molecular Dynamics simulations (RIKEN-Yokohama)

PC Cluster with with MD-Grape-3 LSI chip that calculates the non-bonding 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.

In parallel MDM uses Grape chip to calculate Molecular Dynamics at speed of 100 Tflops.


……

……

…………

MPP for Continuum Simulation(CP-PACS)

Hybrid SystemCommunication Cluster

(Compaq Alpha)

MPP for Particle Simulation(GRAPE-6)

・・

32bit PCI× 8

Paralel File Server(SGI Origin2000)

Parallel Visualization Server

(SGI Onyx2)

100base-TXSwitches

Parallel I/O SystemPAVEMENT/PIO

Parallel Visualization SystemPAVEMENT/VIZ

16 links16 links ・・・・

HMCS for Next-Generation HPC simulations:

(J. Makino)


Magnetic Phenomena in Spherical Tokamak

PurposePurpose ：： IRE (Internal Reconnection Event) IRE (Internal Reconnection Event)

observed in ST (Sherical tokamak) observed in ST (Sherical tokamak) was first simulated by MHD code. was first simulated by MHD code.

Main resultsMain results ：：１．１． Many characteristics observed Many characteristics observed

in ST (time scale of disruption, in ST (time scale of disruption, strange configuration change, strange configuration change, recovery of original spherical recovery of original spherical configuration, etc)configuration, etc)

２．２． Identify the trigger mode (ideal Identify the trigger mode (ideal interchange instability)interchange instability)

３３ . Mechanism of nonlinear . Mechanism of nonlinear deformationdeformation


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

for simulating the ITER plasma within one day.


Heisenberg Uncertainty Relations

• 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)

2. The more precisely I know the time of an event, the less well I know the energy. (and vice versa)

€

Δx • Δp = h

€

ΔE • Δt = h

Werner

Heisenberg


• 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 approxi-mate a rectangular wave packet with length L I need an infinite number of sine waves.

http://www.physics.nwu.edu/ugrad/vpl/waves/index.html

• This bring together a wide range of 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) = Δ x ΔThis is the velocity of the particle.

Wave Packet and its Velocity

v

L


Counting wave length and frequency

• The momentum is related to the wavelength : p = h/

• Δp ~ Δ /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.

• The energy is related to the frequency:• E = h ΔE ~ Δt• In order to determine the frequency with high

accuracy, I have to count wave tops for a very long time.

Δ

• Note that h appears in all these equations. Thus these are quantum mechanical effects.

• http://www.onr.navy.mil/focus/ocean/motion/waves1.htm


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 Δp x L = h• m Δv = h/L

•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

• For the atom L ~ 0.01 nm

• This compares to 13.6 eV for the lowest binding energy of the electron in the hydrogen atom.

• The box size already determines E

L

v

€

ΔE =h2c 2

2L2mc 2~ 40eV


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

• Neutrons and protons are have masses of about 938 MeV and behave inside the box otherwise like electrons.

• Thus photon transitions in nuclei are expected to have a million times higher energy, or frequency.

€

ΔE =(hc)2

2L2mc 2≈1MeV


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 short-range 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.

• 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.


How are nucleons stacked inside a nucleus ?

• Despite the short range interaction 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.

The combined interaction of all nucleons is approximated by the harmonic oscillator well


Heisenberg and particle decay

• 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 ?

Δ

• A neutron is heavier than a proton 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.

€

ΔE = Δ(mc 2) =h

Δt=

6.6 • 10−10eVμs

2.3μs

≈ 3• 10−10eV


How do nuclei and particles decay?

• The Radioactive Decay law says: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

• For Pu 240 T1/2 = 6500 yrs

• For every ton available today 1/2 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

€

dN

dt= −λ • N;

T1/ 2=0.63

λ


Third Homework Set, due Feb. 17, 2005

1. 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?

2. Why does Mr. Pauli allow us to place only 2 electrons into the lowest atomic orbit?

3. How does one produce a wave packet out of waves? Is the speed of the wave packet different from c? More? Less?

4. Why can an electron not drop to the bottom of a box and rest there?5. Use of an electron beam, rather than a light beam, for microscopy allows to

see much smaller (finer) details. Why is that?6. 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.


How to submit Homework

• You have 3 possibilities:

1. Submit it to me in class on the date it is due.

2. Put it in the TA’s (Xiao Shen) mailbox in the Physics Department main office on or before the due date

3. Submit it him by e-mail at the address: [email protected]

• Please DO NOT submit it to me by e-mail

Peter Paul 02/10/05PHY313-CEI544 Spring-051 PHY313 - CEI544 The Mystery of Matter From Quarks to the...

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