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http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes can be found at:
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Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Jan 04, 2016

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Page 1: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

http://crop.unl.edu/claes/PHYS926Lectures.html

The tentative schedule of lectures for the semesterwith links to posted electronic versions of my notes

can be found at:

Page 2: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Introduction to Elementary Particles, David Griffiths, Harper & Row (1987).

Introduction to High Energy Physics, Donald Perkins, Addison-Wesley Publishing.

The Fundamental Particles and Their Interactions, 1st Edition, William B. Rolnick

Page 3: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Two waves of slightly different wavelength and frequency produce beats.

x

x

1k

k = 2

NOTE: The spatial distribution depends on the particular frequencies involved

Page 4: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Many waves of slightly different wavelength can produce “wave packets.”

Page 5: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Adding together many frequencies that are bunched closely together

…better yet…

integrating over a range of frequencies

forms a tightly defined, concentrated “wave packet”

http://phys.educ.ksu.edu/vqm/html/wpe.html

A staccato blast from a whistle cannot be formed by a single pure frequencybut a composite of many frequencies close to the average (note) you recognize

You can try building wave packets at

Page 6: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

The broader the spectrum offrequencies (or wave number)

…the shorterthe wave packet!

The narrower the spectrum offrequencies (or wave number)

…the longerthe wave packet!

Page 7: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Fourier Transforms Generalization of ordinary “Fourier expansion” or “Fourier series”

de)(g2

1)t(f ti

tdetf2

1g tiω)()(

Note how this pairs “canonically conjugate” variables and t.

Whose product must be dimensionless (otherwise eit makes no sense!)

Page 8: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Conjugate variablestime & frequency: t,

f

2

2

What about

coordinate position & ???? r or x

inverse distance??wave number,

p2

In fact through the deBroglie relation, you can write:ph /

/iEte

/hchEc

hhp

/x

ixpe

Page 9: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

x0

For a well-localized particle (i.e., one with a precisely known position at x = x0 )

we could write:

)(0

xx

Dirac -function

a near discontinuous spike at x=x0,(essentially zero everywhere except x=x0 )

x0

with

1)(

0dxxx

such that

dxxxxfdxxxxf )()()()(

000

f(x)≈ f(x0), ≈constant over xx, x+x

x

1x

Page 10: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

For a well-localized particle (i.e., one with a precisely known position at x = x0 )

we could write: )(0

xx

In Quantum Mechanics we learn that the spatial wave function (x) can be complemented by the momentum spectrum of the state, found through the Fourier transform:

dxexp ixp /)(

2

1)(

Here that’s

//

00

2

1)(

2

1)(

pixixp edxexxp

Notice that the probability of measuring any single momentum value, p, is:

2

1

2

1)(

//2

2 00

pixpixeep

What’sTHAT mean?

The probability is CONSTANT – equal for ALL momenta! All momenta equally likely!The isolated, perfectly localized single packet must be comprised of an infinite range of momenta!

Page 11: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

(k) (x)

11

2

kx

k0

2

2

2

kx

(x)(k)

k0

Remember:

…and, recall, even the most general whether confined by some potential OR free actually has some spatial spread within some range of boundaries!

Page 12: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Fourier transforms do allow an explicit “closed” analytic form for

the Dirac delta function

de2

1)t( )t(i

Page 13: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Area within1 68.26%1.28 80.00% 1.64 90.00%1.96 95.00%2 95.44%2.58 99.00%3 99.46%4 99.99%

-2 -1 +1 +2

2

2

2

)x(

e2

1x

Let’s assume a wave packet tailored to be something like aGaussian (or “Normal”) distribution

A single “damped”pulse bounded tightlywithin a few of its

mean postion, μ.

Page 14: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

For well-behaved (continuous) functions (bounded at infiinity)

like f(x)=e-x2/22

dxexfkF ikx)(2

1)(

Starting from:

f(x) g'(x) g(x)= e+ikxik

dxxgx'fxgxf )()()()(

2

1

dxek

ix'fe

k

xif ikxikx )()(

2

1

f(x) is

boundedoscillates in thecomplex plane

over-all amplitude is damped at ±

we can integrate this “by parts”

Page 15: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

dxex'fk

ikF ikx)(

2

1)(

)()(2

1kikFdxex'f ikx

Similarly, starting from:

dkekFxf ikx)(2

1)(

)()(2

1xixfdkek'F ikx

Page 16: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

And so, specifically for a normal distribution: f(x)=ex2/22

differentiating: )()(2

xfx

xfdx

d

from the relation just derived: kdekF

ixf

dx

d xki ~)

~(

2

1)(

~

2 '

Let’s Fourier transform THIS statement

i.e., apply: dxeikx

2

1on both sides!

dxei

kikF ikx 2

1)( 2 1

2 F'(k)e-ikxdk~ ~~

kdkFi ~

)~

(2 '

ei(k-k)xdx~ 1

2

(k – k)~

Page 17: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

kdkFi

kikF~

)~

()( 2 '

ei(k-k)xdx~ 1

2

(k – k)~

)()( 2 kFi

kikF '

selecting out k=k~

rewriting as: 2

)(

/)( kkF

dkkdF

0

k

0

k

dk''

''dk'

22

2

1)0(ln)(ln kFkF

2221

)0(

)( ke

F

kF 22

21

)0()(k

eFkF

Page 18: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

2221

)0()(k

eFkF

22 2/)( xexf Fourier transforms

of one anotherGaussian distribution

about the origin

dxexfkF ikx)(2

1)(

Now, since:

dxxfF )(2

1)0(

we expect:

10 xie

22

1)0(

22 2/

dxeF x

2221

2)(k

ekF

22 2/)( xexf Both are of the form of a Gaussian!

x k 1/

Page 19: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

x k 1

orgiving physical interpretation to the new variable

x px h

Page 20: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

xk ~ 2

t ~ 2

xp ~ h

tE ~ h

Page 21: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

To be charged: means the particle is capable of emitting and absorbing photons

What’s the ground state or zero-point energy of a system?

e

e

harmonic oscillator: ½h

Page 22: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

xp ~ h tE ~ h

The virtual photons in the sea that surround every charged particle, are wavepackets centered at the origin (source of charge)

If these describe/map out the electrostatic potentialand

relate available momentum to be transferredto the distance from the field’s source

consider the extremes:

x 0

x

other charges may be exposed to thefull spectrum of possible momenta

only vanishingly small momentum transfers are possible

Page 23: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1896

1899

1912

Page 24: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Henri Becquerel (1852-1908) received the 1903 Nobel Prize in Physics for the discovery of natural radioactivity.

Wrapped photographic plate showed clear silhouettes, when developed, of the uranium salt samples stored atop it.

1896 While studying the photographic images of various fluorescent & phosphorescent materials, Becquerel finds potassium-uranyl sulfate spontaneously emits radiation capable of penetrating thick opaque black paper

aluminum plates copper plates

Exhibited by all known compounds of uranium (phosphorescent or not) and metallic uranium itself.

Page 25: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 26: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1898 Marie Curie discovers thorium (90Th) Together Pierre and Marie Curie discover polonium (84Po) and radium (88Ra)

1899 Ernest Rutherford identifies 2 distinct kinds of rays emitted by uranium - highly ionizing, but completely

absorbed by 0.006 cm aluminum foil or a few cm of air

- less ionizing, but penetrate many meters of air or up to a cm of

aluminum.

1900 P. Villard finds in addition to rays, radium emits - the least ionizing, but capable of penetrating many cm of lead, several feet of concrete

Page 27: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 28: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

B-fieldpoints

into page

1900-01 Studying the deflection of these rays in magnetic fields, Becquerel and the Curies establish rays to be charged particles

Page 29: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1900-01 Using the procedure developed by J.J. Thomson in 1887 Becquerel determined the ratio of charge q to mass m for

: q/m = 1.76×1011 coulombs/kilogram identical to the electron!

: q/m = 4.8×107 coulombs/kilogram 4000 times smaller!

Page 30: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Discharge Tube

Thin-walled(0.01 mm)glass tube

to vacuumpump &Mercurysupply

Radium or Radon gas

Noting helium gas often found trapped in samples of radioactive minerals, Rutherford speculated that particles might be doubly ionized Helium atoms (He++)

1906-1909 Rutherford and T.D.Royds develop their “alpha mousetrap” to collect alpha particles and show this yields a gas with the spectral emission lines of helium!

Mercury

Page 31: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Status of particle physics early 20th century

Electron J.J.Thomson 1898

nucleus ( proton) Ernest Rutherford 1908-09

Henri Becquerel 1896 Ernest Rutherford 1899

P. Villard 1900

X-rays Wilhelm Roentgen 1895

Page 32: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 33: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1900 Charles T. R. Wilson’s ionization chamber Electroscopes eventually discharge even when all known causes are removed, i.e., even when electroscopes are

•sealed airtight•flushed with dry,

dust-free filtered air•far removed from any

radioactive samples •shielded with 2 inches of lead!

seemed to indicate an unknown radiation with greater

penetrability than x-rays or radioactive rays

Speculating they might be extraterrestrial, Wilson ran underground tests at night in the Scottish railway, but

observed no change in the discharging rate.

Page 34: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1909 Jesuit priest, Father Thomas Wulf , improved the ionization chamber with a design planned specifically for high altitude balloon flights.

A taut wire pair replaced the gold leaf.

This basic design became the pocket dosimeter carried to record one’s total exposure to ionizing radiation.

0

Page 35: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1909 Taking his ionization chamber first to the top of the Eiffel Tower (275 m) Wulf observed a 64% drop in the discharge rate.

Familiar with the penetrability of radioactive rays, Wulf expected any ionizing effects due to natural radiation from the ground, would have been heavily absorbed by the “shielding” layers of air.

Page 36: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

•light produces spots of submicroscopic silver grains•a fast charged particle can leave a trail of Ag grains

•1/1000 mm (1/25000 in) diameter grains

•small singly charged particles - thin discontinuous wiggles•only single grains thick

•heavy, multiply-charged particles - thick, straight tracks

1930s plates coated with thick photographic emulsions (gelatins carrying silver bromide crystals) carried up mountains or in balloons clearly trace cosmic ray tracks through their depth when developed

Page 37: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

November 1935 Eastman Kodak plates

carried aboard Explorer II’s record altitude

(72,395 ft) manned flight into

the stratosphere

Page 38: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

50m

Cosmic ray strikes a nucleuswithin a layer of

photographicemulsion

1937 Marietta Blau andHerta Wambacher

report “stars” of tracks resulting from cosmic

ray collisions with nuclei within the emulsion

Page 39: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Elastic collision

Page 40: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

p

p

p

p p

p

Page 41: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1894 After weeks in the Ben Nevis Observatory, British Isles, Charles T. R. Wilsonbegins study of cloud formation

•a test chamber forces trapped moist air to expand•supersaturated with water vapor•condenses into a fine mist upon the dust particles in the air

each cycle carried dust that settled to the bottom

purer air required larger, more sudden expansion observed small wispy trails of droplets forming without dust to condense on!

Page 42: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 43: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Tracks from an alpha source

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Page 47: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1952 Donald A. Glaser invents the bubble chamber

•boiling begins at nucleation centers (impurities) in a volume of liquid

•along ion trails left by the passage of charged particles•in a superheated liquid tiny bubbles form for ~10 msec before obscured by a rapid, agitated “rolling” boil

•hydrogen, deuterium, propane(C3H6) or Freon(CF3Br) is stored as a liquid at its boiling point by external pressure (5-20 atm)•super-heated by sudden expansion created by piston or diaphragm•bright flash illumination and stereo cameras record 3 images through the depth of the chamber (~6m resolution possible)

•a strong (2-3.5 tesla) magnetic field can identify the sign of a particle’s charge and its momentum (by the radius of its path)

1960 Glaser awarded the Nobel Prize for Physics

Page 48: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 49: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

3.7m diameter Big European Bubble Chamber

CERN (Geneva, Switzerland)

Side View

Top View

Page 50: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 51: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
Page 52: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1936 Millikan’s group shows at earth’s surface cosmic ray showers are dominated by electrons, gammas, and X-particles capable of penetrating deep underground (to lake bottom and deep tunnel experiments) and yielding isolated single cloud chamber tracks

Page 53: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1937 Street and Stevenson1938 Anderson and Neddermeyer determine X-particles

•are charged•have 206× the electron’s mass•decay to electrons with a mean lifetime of 2sec

0.000002 sec

Page 54: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Schrödinger’s Equation

Based on the constant (conserved) value of the Hamiltonian expression

EVpm

2

2

1 total energy sum of KE + PE

with the replacement of variables by “operators”

t

iVm

22

2

i

p

tiE

As enormously powerful and successful as this equation is,what are its flaws? Its limitations?

Page 55: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

We could attempt a RELATIVISTIC FORM of Schrödinger:

What is the relativistic expression for energy?

42222 cmcpE relativistic energy-momentum relation

2

222

2

2

2

1

cm

tc

As you’ll appreciate LATER

this simple form (devoid of spin factors)describes spin-less (scalar) bosons

For m=0 this yields the homogeneous differential equation:

01

2

2

2

2

tc

Which you solved in E&M to find that wave equations forthese fields were possible (electromagnetic radiation).

Page 56: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

(1935) Hideki Yukawa saw the inhomogeneous equation as possibly descriptive of a scalar particle mediating SHORT-RANGE forces

like the “strong” nuclear force between nucleons (ineffective much beyond the typical 10-15 meter

extent of a nucleus

2

222

2

2

2

1

cm

tc

For a static potential drop 2

2

t

and assuming a spherically symmetric potential, can cast this equation in the form:

)()(1

)(2

222

2

2 rUcm

r

Ur

rrrU

with a solution (you will verify for homework):

Rrer

grU /

4)(

where R=

hmc

Page 57: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Rrer

grU /

4)(

where R=

hmcLet’s compare:

to the potential of electromagnetic fields: r

grU

4)(

with e-r/R=1its like Ror m = 0!

For a range something like 10-15 mYukawa hypothesized the existence

of a new (spinless) boson with mc2 ~ 100+ MeV.

In 1947 the spin 0 pion was identified with a mass ~140 MeV/c2

Page 58: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

1947 Lattes, Muirhead, Occhialini and Powell observe pion decay

Cecil Powell (1947)Bristol University

Page 59: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

C.F.Powell, P.H. Fowler, D.H.PerkinsNature 159, 694 (1947) Nature 163, 82 (1949)

Page 60: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Quantum Field TheoryNot only is energy & momentum QUANTIZED (energy levels/orbitals)

but like photons are quanta of electromagnetic energy,all particle states are the physical manifestation of quantummechanical wave functions (fields).

Not only does each atomic electron exist trapped within quantized energy levels or spin states,

but its mass, its physical existence, is a quantum state of a matter field.

e

the quanta of the em potential virtual photonsas opposed to observable photons

These are not physical photons in orbitals about the electron. They are continuouslyand spontaneously being emitted/reabsorbed.

Page 61: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

The Boson PropagatorWhat is the momentum spectrum of Yukawa’s massive (spin 0) relativistic boson?

Remember it was proposed in analogy to the E&M wave functions of a photon.What distribution of momentum (available to transfer) does a

quantum wave packet of this potential field carry?

dVerUqf rqi

)(

2

1)(

3

q r = qrcos

dV = r2 d sin d drIntegrating the angular part:

drdedrUrqf iqr

sin)(

2

1)(

0

2

0

2 cos3

2iqr

eee

iqr

iqriqriqr

0

cos1

drrqr

qrrU 2

0

sin)(4

2

12/3

22

2/321)(

mq

gqf

The more massive the mediating boson,the smaller this distribution…

Page 62: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Consistently ~600 microns (0.6 mm)

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Page 64: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.
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pdg.lbl.gov/pdgmail

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BraKet notation We generalize the definitions of vectors and inner products ("dot" products) to extend the formalism to functions (like QM wavefunctions) and differential operators.

v = vx x + vy y + vz z n vn n

then the inner product is denoted by

v u =

^ ^ ^ ^

n vn un

sometimes represented by row and column matrices:

[vx vy vz ] ux

uy = [ ]

uz

vxux + vyuy + vzuz

Remember: n m = nm ^ ^

Page 73: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

We most often think of "vectors" in ordinary 3-dim space, but can immediately and easily generalize to COMPLEX numbers:

v u = n

[vx vy vz ] ux

uy = [ ]

uz

n vn*

un

vx*ux + vy

*uy + vz*uz

and by the requirement

< v | u > = < v | u >*we guarantee that the “dot product” is real

transpose column into row and take complex conjugate

* * *

Page 74: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Every “vector” is a ket : |v1> |v2>including the unit “basis” vectors.

We write: | v > = n | >

and the scalar product by the symbol

< | >

and the orthonormal condition on basis vectors can be stated as

< | > = Now if we write

| v1 > = C1n|n> and | v2 > = C2

n|n> then

“we know”:

< v2 | v1 > = nC2n* C1

n =

because of orthonormality

< v2 | | v1 > = m

“bra”

Cn n

v u

m n mn

n,mC2m

* C1n<m|n>

mC2m

* <m|mC1n|n>

Page 75: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

So if we write | v > = Cn|n> = n |n>

= n

= {n } =

So what should this give? < n | v1 > = ??

Remember: < m | n > gives a single element 1 x 1 matrix but: | m > < n | gives a ???

C1n

<n|v>

|n><n|v>

| v > |n><n| |v>1 |v>

Page 76: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

n|n><n|

In the case of ordinary 3-dim vectors, this is a sum over the products:

100

[ 1 0 0 ] 010

[ 0 1 0 ] 001

[ 0 0 1 ]+ +

1 0 00 0 00 0 0

+=0 0 00 1 00 0 0

+0 0 00 0 00 0 1

1 0 00 1 00 0 1

=

Page 77: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

e

Two important BASIC CONCEPTS

•The “coupling” of a fermion (fundamental constituent of matter)

to a vector boson (the carrier or intermediary of interactions)

•Recognized symmetries are intimately related to CONSERVED quantities in nature which fix the QUANTUM numbers describing quantum states and help us characterize the basic, fundamental interactions between particles

Page 78: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

Should the selected orientation of the x-axis matter?

As far as the form of the equations of motion? (all derivable from a Lagrangian)

As far as the predictions those equations make?Any calculable quantities/outcpome/results?

Should the selected position of the coordinate origin matter?

If it “doesn’t matter” then we have a symmetry: the x-axis can be rotated through any direction of 3-dimensional space

orslid around to any arbitrary location

and the basic form of the equations…and, more importantly, all thepredictions of those equations are unaffected.

Page 79: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

If a coordinate axis’ orientation or origin’s exact location “doesn’t matter” then it shouldn’t appear explicitly in the Lagrangian!

EXAMPLE: TRANSLATION

Moving every position (vector) in space by a fixed a(equivalent to “dropping the origin back” –a)

original descriptionof position

r

–a

r' new descriptionof position

ar'r

iii qq

r'r

dq

rd

'

a

a

i

iii

i dq

qrdq(qr

dq

rd )() a

dq

adq

i

i ˆˆ

or

Page 80: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

For a system of particles:

N

iirmT

1

2

21

acted on only by CENTAL FORCES: )()( rVrV function of separation

0

kk q

L

q

L

dt

d

no forces externalto the system

generalized momentum(for a system of particles,

this is just the ordinary momentum)

kk ppdt

d kk q

V

q

L

=for a system of particles

T may depend on q or r

but never explicitly on qi or ri

k

i

ii

k q

r

r

Vp

Page 81: Http://crop.unl.edu/claes/PHYS926Lectures.html The tentative schedule of lectures for the semester with links to posted electronic versions of my notes.

For a system of particles acted on only by CENTAL FORCES:

k

i

ii

k q

r

r

Vp

-Fi a

aFpi

ik ˆ

aFtotal ˆ

net force on a systemexperiencing only

internal forcesguaranteed

by the 3rd Lawto be

0 kp

Momentummust be conservedalong any direction

the Lagrangian is invariant totranslations in.