Transcript Yukawa`s Pion, Bra Ket Notation - Cosmic Ray Observatory Project

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.
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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.
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
50mm
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 (~6mm
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
Primary proton
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 2msec
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   0
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
4 

0
1 iqrcos

e
iqr
1/ 2
 sin qr 2
2
U (r )
r dr   
qr
 
1 2 3/ 2 g

f (q)   2
q  m2

0
0
e iqr  eiqr

iqr
g mcr /  sin qr 2
e
r dr
4r
qr
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 > =
and
m| n >
= 
mn
| v2 > = SC2n|n> then
SnC2n* C1n
=
SSn,mC2m* C1n<m|n>
because of orthonormality
< v2 | | v1 > =
“bra”
SmC2m* <m|SnC1n|n>
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.

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