Download Basic Ideas for Particle Properties

Basic Ideas for Particle
Properties
Review of Modern Physics
Spin and Angular Momentum

There are three angular momenta.



Orbital (“normal” angular momentum)
Spin (“intrinsic” angular momentum)
Total (orbital + spin)
Orbital angular momentum:
  
Lrp
 
 

Lˆ z  i x  y   i
x 

 y
(Classical)
(Quantum)

Quantum mechanical
properties

Eigenvalue equations
L2 lm    1 2 lm
Lz lm  m lm
S 2 s , s z  34  2 s , s z
J 2 JM  j  j  1 2 JM
J z JM  M JM

Commutation relations
L , L   
J , J   
iLk
i
j
ijk
i
j
ijk
iJ k

The eigenvalues are quantized.


The appropriate unit for spin is .
The only quantities are:


0, 1/2, 1, 3/2, 2, 5/2, …
There are only 2J+1 possible values.
Fermions and Bosons
Example
J
Quantum
statistics
Many body
wave function
Fermions
Nucleons,
electron,
quarks, etc.
Half-integer
Only one
fermion per
state (Pauli’s
exclusion
principle
Antisymmetric
Bosons
Photon,
W, Z, gluon,
, etc.
Integer
Any number
of identical
particles
capable of
occupying the
same state
Symmetric

The wave functions for
two-body system
 1,2    2,1 symmetric
 1,2   2,1 anti - symmetric

What are the practical
wave functions?
 1,2   A 1 2  B 2 1 symmetric
 1,2   A 1 2  B 2 1 anti - symmetric
Magnetic Dipole Moment (Magneton)

Particles having spin can get the magnetic energy.

The constant is called magnetic dipole moment, .


The magnetic moment is dependent on the spin,
mass and charge.
The intrinsic constant in the  is called magneton,
0=e/2mc.
Mass Measurements 1

Mass spectroscopy


This utilizes the centripetal and Lorentz forces to
find out the particle mass.
This is useful for nuclei and atoms, but it is
impossible for most particles.


The initial velocity of particle produced by reaction
cannot be known exactly.
Neutral charges are not deflected by a magnetic field.
Mass Measurements 2

Scintillation counter

This utilizes two scintillation counters to measure
the velocity of a particle.




Magnet selects particles with momentum.
Two counters and oscilloscope measure the distance
and time to give the velocity.
The mass is the above momentum divided by the
velocity.
This method fails if the particle is neutral and the
life time is very short.
Mass Measurements 3

Invariant mass plot


This utilizes the
invariant mass of
particles.
Let’s suppose you
measure the mass of
neutral rho, 0 (the life
time is 610-24 sec). It
decays into + and -.
+
-
E1 p1
E2 p2
E p
0
Mass Measurements 3 (cont.)

Invariant mass for the
pions


1
2
2 2 12
m12  2 E1  E2   p1  p 2  c
c

Energy & momentum
for rho
E  E1  E2 , p   p1  p 2

Invariant mass for the
rho
m 


1 2
2 2 12
E

p

c
2
c
Namely, m = m12.
Mass Measurement 3 (cont.)

The invariant mass plot is capable of
measuring mass of very-short-life-time
particles.

Not only elementary particles, but it is used
for nuclear physics region. (e.g. 8Be)
Particles and the Related Interactions

Forces & particle interactions

Remember four forces.

But those are pure form of interaction.

In practice, the interactions between particles
are mixed.
Particles
Type
Weak
Electromagnetic
Hadronic
Photon
Gauge boson
No
Yes
No
W Z0
Gauge boson
Yes
Yes
No
Gluon
Gauge boson
No
No
Yes
Neutrino
Fermion
Yes
No
No
Electron
Fermion
Yes
Yes
No
Muon
Fermion
Yes
Yes
No
Mesons
Bosons
Yes
Yes
Yes
Baryons
Fermions
Yes
Yes
Yes
Quarks
Fermions
Yes
Yes
Yes
Leptons
Hadrons
Decays

Phenomenological
point of view

The number decaying
in a time dt 
dN  N t  dt

The number of particles
present at time t 
N t   N 0e
 t
Decays (cont.)




Write it in terms of the
time-dependent wave
function.
But it doesn’t work!
So introduce the
imaginary part. 
Then, that makes sense.
 iEt 
 t    0 exp  

  
E  E0  12 i
 iE 0t 
 t 
 t    0 exp  
 exp   
 2 
  
Decay (cont.)


However, what is the ? Is there any physical
meaning?
Now let’s transform it into the “energy
space.”


The wave function will be expressed in terms of energy instead of
time. The modulus square of the function will be the probability
density.
It turns out that the  is the uncertainty of
energy at a decaying state.

In other words, the  is the full width at half maximum.
Decays (cont.)

Possibility of the decay properties and its
classification…



There is no simple connection between decay appears and
other particle properties…
Decay energies differ with hadronic, electromagnetic and
weak forces.
The output from interaction and the decay time are not
related. (It involves a deeper rule.)