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:

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

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

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

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

The broader the spectrum offrequencies (or wave number)

…the shorterthe wave packet!

The narrower the spectrum offrequencies (or wave number)

…the longerthe wave packet!

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!)

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

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

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!

(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!

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

the Dirac delta function

de2

1)t( )t(i

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.

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”

dxex'fk

ikF ikx)(

2

1)(

)()(2

1kikFdxex'f ikx

Similarly, starting from:

dkekFxf ikx)(2

1)(

)()(2

1xixfdkek'F ikx

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)~

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

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/

x k 1

orgiving physical interpretation to the new variable

x px h

xk ~ 2

t ~ 2

xp ~ h

tE ~ h

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

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

1896

1899

1912

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.

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

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

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!

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

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

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.

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

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.

•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

November 1935 Eastman Kodak plates

carried aboard Explorer II’s record altitude

(72,395 ft) manned flight into

the stratosphere

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

Elastic collision

p

p

p

p p

p

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!

Tracks from an alpha source

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

3.7m diameter Big European Bubble Chamber

CERN (Geneva, Switzerland)

Side View

Top View

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

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

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?

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).

(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

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

1947 Lattes, Muirhead, Occhialini and Powell observe pion decay

Cecil Powell (1947)Bristol University

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

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.

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…

Consistently ~600 microns (0.6 mm)

pdg.lbl.gov/pdgmail

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 ^ ^

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

* * *

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>

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>

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

=

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

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.

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

aˆ

i

iii

i dq

qrdq(qr

dq

rd )() a

dq

adq

i

i ˆˆ

or

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

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.