The tentative schedule of lectures for the semester
with links to posted electronic versions of my notes
can be found at:
http://crop.unl.edu/claes/PHYS926Lectures.html
Introduction to Elementary Particles,
David Griffiths, Harper & Row (1987).
The Fundamental Particles
and Their Interactions,
1st Edition, William B. Rolnick
Introduction to High Energy Physics,
Donald Perkins, Addison-Wesley Publishing.
NOTE: The spatial distribution depends on the particular frequencies involved
x
1
x 
k
k =
2

Two waves of slightly different wavelength and frequency produce beats.
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”
A staccato blast from a whistle cannot be formed by a single pure frequency
but a composite of many frequencies close to the average (note) you recognize
You can try building wave packets at
http://phys.educ.ksu.edu/vqm/html/wpe.html
The broader the spectrum of
frequencies (or wave number)
…the shorter
the wave packet!
The narrower the spectrum of
frequencies (or wave number)
…the longer
the wave packet!
Fourier Transforms
Generalization of ordinary “Fourier expansion” or “Fourier series”

f (t) 
1
 it
g
(

)
e
d


2  
1
g( ) 
2

f
(
t
)
e

 iωt
dt

Note how this pairs “canonically conjugate” variables  and t.
Whose product must be dimensionless (otherwise eit makes no sense!)
Conjugate variables
time & frequency:
t, 
  2
 2 f
What about
coordinate position & ????
inverse distance??
2
r or x

wave number,  
In fact through the deBroglie relation,
E  h  hc /   
e
iEt / 
  h/ p
h
p 
 c
h
ixpx / 
e

you can write:
p

For a well-localized particle (i.e., one with a precisely known position at x = x0 )
we could write:
   ( x  x0 )
Dirac -function
a near discontinuous spike at x=x0,
(essentially zero everywhere except x=x0 )

with
  ( x  x0 )dx  1
1
x
x0
x
x0

such that

 f ( x) ( x  x0 )dx  f ( x0 )  ( x  x0 )dx
f(x)≈ f(x0), ≈constant over xx, x+x
For a well-localized particle (i.e., one with a precisely known position at x = x0 )
we could write:
0
   (x  x )
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:
1
 ( p) 
2
Here that’s
1
 ( p) 
2



 ( x )e
  ( x  x0 )e
ixp / 
ixp / 
dx
1 ix0 p / 
dx 
e
2
Notice that the probability of measuring any single momentum value, p, is:
2
 1  ix0 p /  ix0 p /  1
 ( p)  
e

 e
2
 2 
2
What’s
THAT 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!
Remember:
(k)
(x)
k0
x1 
2
k1
(x)
(k)
k0
x2 
…and, recall, even the most general  whether confined by some potential
OR free actually has some spatial spread within some range of boundaries!
2
k2
Fourier transforms
do allow an explicit “closed” analytic form for
the Dirac delta function
1
 (t   ) 
2
  i ( t  )
e
d



Let’s assume a wave packet tailored to be something like a
Gaussian (or “Normal”) distribution
Area within
1
1.28
1.64
1.96
2
2.58
3
4
68.26%
80.00%
90.00%
95.00%
95.44%
99.00%
99.46%
99.99%
A single “damped”
pulse bounded tightly
within a few  of its
mean postion, μ.
-2
-1

+1
+2
1
 x  
e
 2
( x   )2

2
2
For well-behaved (continuous) functions (bounded at infiinity)
2/22
-x
like f(x)=e

Starting from:
1
ikx
F (k ) 
f ( x )e dx

2 
we can integrate this “by parts” f(x)
g'(x)
g(x)=
i +ikx
e
k

1 


f ( x ) g ( x )    f ' ( x ) g ( x )dx

2 




1  if ( x ) ik x
i ik x 

e
  f ' ( x ) e dx 

2  k
k



f(x) is
bounded
oscillates in the
complex plane
over-all amplitude is damped at ±
i 1
F (k ) 
k 2
1
2

 f ' ( x )e


 f ' ( x )e
ikx
dx
ikx
dx  ikF (k )

Similarly, starting from:
1
f ( x) 
2

 F ( k )e
ikx
dk

1
2

 F' ( k ) e

ikx
dk  ixf ( x )
And so, specifically for a normal distribution: f(x)=ex
2/22
d
x
f ( x)   2 f ( x)
dx

d
i 1
~ ik~x ~
f ( x)  2
F' (k )e dk

dx
 2
differentiating:
from the relation
just derived:
Let’s Fourier transform THIS statement
1
i.e., apply:
2

 
eikxdx
 ikF (k ) 
i


2
i

2
on both sides!
1
ikx
1
~
~
~
-ikx
F'(k)e dk e dx
2 2
~ ~
~
1 ei(k-k)x
dx F' (k )dk
2
~
 (k – k)
 ikF (k ) 
i

2
~ ~
~
1 ei(k-k)x
dx F' (k )dk
2
~
 (k – k)
 ikF (k ) 
i

F' (k )
2
~
selecting out k=k
k
rewriting as:
dF ( k' ) / dk'
dk'
F ( k')
0
k
  k dk'
2
0
1 2 2
ln F ( k )  ln F (0)    k
2
 1 2k 2
F (k )
e 2
F (0)
F (k )  F (0)e
1 2k 2
2
f ( x)  e
 x2 / 2 2
Fourier transforms
of one another
F (k )  F (0)e
1 2k 2
2
Gaussian distribution
about the origin
Now, since:
we expect:
1 
ikx
F (k ) 
f
(
x
)
e
dx

2 
i0 x
e
1

1
F (0) 
f ( x )dx

2 
1   x2 / 2 2
F (0) 
e
dx  2 

2 
f ( x)  e
 x2 / 2 2
x  
Both are of the form
of a Gaussian!
F (k )  2e
k  1/
1 2k 2
2
x k  1
or
giving physical interpretation to the new variable
x px  h
xk ~ 2
xp ~ h
t ~ 2
tE ~ h
To be charged: means the particle is capable of emitting and absorbing photons

e

e
What’s the ground state or zero-point energy of a system?
harmonic oscillator: ½h
The virtual photons in the sea that surround every charged particle,
are wavepackets centered at the origin (source of charge)
xp ~ h
tE ~ h
If these describe/map out the electrostatic potential
and
relate available momentum to be transferred
to the distance from the field’s source
consider the extremes:
x  0
other charges may be exposed to the
full spectrum of possible momenta
x  
only vanishingly small momentum
transfers are possible
1896
1899
1912
a, b

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
a - highly ionizing, but completely
absorbed by 0.006 cm aluminum
foil or a few cm of air
b - less ionizing, but penetrate many
meters of air or up to a cm of
aluminum.
1900 P. Villard finds in addition to a rays, radium emits  - the least ionizing,
but capable of penetrating many cm of lead, several feet of concrete
B-field
points
into page
1900-01 Studying the deflection of these rays in magnetic fields,
Becquerel and the Curies establish a, b 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
b: q/m = 1.76×1011 coulombs/kilogram
identical to the electron!
a: q/m = 4.8×107 coulombs/kilogram
4000 times smaller!
Discharge Tube
Thin-walled
(0.01 mm)
glass tube
Noting helium gas often found trapped in samples
of radioactive minerals, Rutherford speculated that
a 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!
to vacuum
pump &
Mercury
supply
Radium or
Radon gas
Mercury
Status of particle physics
early 20th century
Electron
J.J.Thomson
1898
nucleus ( proton) Ernest Rutherford 1908-09
a
Henri Becquerel 1896
Ernest Rutherford 1899
b

P. Villard
X-rays
Wilhelm Roentgen 1895
1900
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
a, b,  rays, Wulf expected any ionizing effects
due to natural radiation from the ground, would
have been heavily absorbed by the “shielding”
layers of air.
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
•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
November 1935
Eastman Kodak plates
carried aboard
Explorer II’s
record altitude
(72,395 ft)
manned flight into
the stratosphere
1937 Marietta Blau and
Herta Wambacher
report “stars” of tracks
resulting from cosmic
ray collisions with
nuclei within the emulsion
50m
Cosmic ray
strikes a nucleus
within a layer of
photographic
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
•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
1952 Donald A. Glaser
invents the
bubble chamber
•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 (~6m
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
Side View
3.7m diameter
Big European Bubble Chamber
CERN
(Geneva, Switzerland)
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 Stevenson
1938 Anderson and Neddermeyer
determine X-particles
•are charged
•have 206× the electron’s mass
•decay to electrons with
a mean lifetime of 2sec
0.000002 sec
Schrödinger’s Equation
Based on the constant (conserved) value of the Hamiltonian expression
1 2
p V  E
2m
total energy  sum of KE + PE
with the replacement of variables by “operators”


p 
i

E  i
t
2




2
  V   i 

t
 2m

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?
E 2  p 2c 2  m2c 4
1 
mc
2
    2 
2
2
c t

2
2
2
relativistic energy-momentum relation
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:
2
1

 2

     2 2   0
c t 

Which you solved in E&M to find that wave equations for
these fields were possible (electromagnetic radiation).
(1935) Hideki Yukawa saw the inhomogeneous
equation as possibly descriptive of a scalar
particle mediating SHORT-RANGE forces
1 
mc
2
    2 
2
2
c t

2
2
2
like the “strong” nuclear force between nucleons
(ineffective much beyond the typical 10-15 meter
extent of a nucleus
2
For a static potential drop 2 
t
and assuming a spherically symmetric potential, can cast this equation in the form:
1  2 U
mc
 U (r)  2 (r
)  2 U (r)
r r
r

2
2
2
with a solution (you will verify for homework):
g r / R
U (r) 
e
4r
where
R=
h
mc
Let’s compare:
g r / R
U (r) 
e
4r
to the potential of
electromagnetic fields:
g
U (r) 
4r
where
R=
h
mc
with e-r/R=1
its like R
or m = 0!
For a range something like 10-15 m
Yukawa 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.Perkins
Nature 159, 694 (1947)
Nature 163, 82 (1949)
Quantum Field Theory
Not only is energy & momentum QUANTIZED (energy levels/orbitals)
but like photons are quanta of electromagnetic energy,
all particle states are the physical manifestation of quantum
mechanical 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.
the quanta of the em potential  virtual photons
as opposed to observable photons
e
These are not physical photons in orbitals
about the electron. They are continuously
and spontaneously being emitted/reabsorbed.
The Boson Propagator
What 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?
3
  1 
 iqr
q r = qrcos
f (q)  
  U ( r )e
 2 
dV
dV = r2 d sin d dr
Integrating the angular part:
3
  1  2  2
 iqr cos
f (q)  
sin  d dr
  r U ( r ) 0 d 0 e
 2 

2
 1 


 2 
3/ 2
0
e iqr  eiqr

iqr
 sin qr 2
4 0 U ( r )
r dr
qr
1 2 3/ 2 g

f (q)   2
q  m2
1 iqrcos

e
iqr

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^ 
Sn vn ^n
then the inner product is denoted by
v u =
Sn vn un
^=
Remember: n^ m
nm
sometimes represented by row and column matrices:
[vx vy vz ] ux
uy = [ vxux + vyuy + vzuz ]
uz
We most often think of "vectors" in ordinary 3-dim space, but
can immediately and easily generalize to COMPLEX numbers:
v u =
Sn vn* un
[vx* vy* vz* ] ux
uy = [ vx*ux + vy*uy + vz*uz ]
uz
transpose column into row
and take complex conjugate
and by the requirement
<v|u>
=
< v | u >*
we guarantee that the “dot product” is real
Every “vector” is a ket :
including the unit “basis” vectors.
|v1>
We write:
|v> =
|v2>
SnCn | n >
and the scalar product by the symbol
< v |u >
and the orthonormal condition on basis vectors can be stated as
<
Now if we write
| v1 > = SC1n|n>
“we know”:
< v2 | v1 > =
< v2 | | v1 > =
“bra”
and
= 
m| n >
mn
| v2 > = SC2n|n> then
SnC2n* C1n
=
Sn,mC2m* C1n<m|n>
SmC2m* <m|SmC1n|n>
because of orthonormality
So what should this give?
C1n
< n | v1 > = ??
So if we write | v > = SCn|n> = Sn<n|v> |n>
= Sn |n><n|v>
| v > = {Sn |n><n| } |v> =
1 |v>
Remember: < m | n > gives a single element 1 x 1 matrix
but: | m > < n | gives a ???
Sn|n><n|
In the case of ordinary 3-dim vectors, this is a sum over the products:
1 [ 1 0 0 ]
0
+
0
=
1 0 0
0 0 0
0 0 0
0 [ 0 1 0 ]
1
+
0
+
0 0 0
0 1 0
0 0 0
=
1 0 0
0 1 0
0 0 1
+
0 [ 0 0 1 ]
0
1
0 0 0
0 0 0
0 0 1
Two important BASIC CONCEPTS
•The “coupling” of a fermion
(fundamental constituent of matter)
to a vector boson
(the carrier or intermediary of interactions)
e
•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
or
slid around to any arbitrary location
and the basic form of the equations…and, more importantly, all the
predictions 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)
–a
  
r'  r  a


 
a
dr
r'  r
  aˆ

dqi q' i qi a
original
description
of position
r
r' new
description
of position
or
 

dr r (qi  dqi )  r ( qi ) dqi aˆ


 aˆ
dqi
dqi
dqi
For a system of particles:
T
N
1
2

m
r
2 i
i 1
acted on only by CENTAL FORCES:
d L L

0
dt qk qk

V (r )  V (r)
function of
separation
no forces external
to the system
generalized momentum
(for a system of particles,
this is just
the ordinary momentum)
d
L
V
pk  p k =

dt
qk
q

V ri
p k  
i r q
i
k
for a system of particles
T may depend on
q or r
but never explicitly
k
on qi or ri
For a system of particles acted on only by CENTAL FORCES:

V ri
p k  
i r q
i
k
-Fi a^


p k   Fi  aˆ  Ftotal  aˆ
i
p k
net force on a system
experiencing only
internal forces
guaranteed
by the 3rd Law Momentum
must be conserved
to be
along any direction
the Lagrangian
is invariant to
translations in.

0