PC4245 PARTICLE PHYSICS
HONOURS YEAR
Tutorial 2
1. Suppose we interpret the electron literally as a classical solid sphere of radius r,
1
mass m, spinning with angular momentum  . What is the speed, v, of a point
2
on its “equator”? Experimentally, it is known that r is less than 10 16 cm. What is
the corresponding equatorial speed? What do you conclude from this?
2. Suppose an electron is in the state
1/ 5
2/ 5
a. If you measured Sx, what values might you get, and what is the probability of
each?
b. If you measured Sy, what values might you get, and what is the probability of
each?
c. If you measured Sz, what values might you get, and what is the probability of
each?
3. a. Show that  x2   y2   z2  1 . (“1” here really means 2 x 2 unit matrix; if no
matrix is specified, the unit matrix is understood).
b.
Show that  x y  i z ,  y z  i x ,  z  x  i y . These results are neatly
summarized in the formula
 i j   ij  i ijk k
(summation over k implied), where  ij is the Kronecker delta:
1, if i = j
 ij 
0, otherwise
and ijk is the Levi-Civita symbol
ijk
1, if ijk = 123, 231 or 312
-1, if ijk = 132, 213 or 321
0, otherwise
4. a. Show that e i z / 2  i z .
b. Find the matrix U representing a rotation by 180  about the y axis, and show
that it coverts “spin up”, into “spin down”, as we would expect.
c. More generally, show that
U ( )  exp( i    / 2)  cos
~

~


2
 i (   ) sin
~
2
~

where  is the magnitude of  , and    / 
~
~
~
5. The  * can decay into      ,  0   0 , or      . Suppose you
observed 100 such disintegrations, how many would you expect to see of each
type ?
0
6. Consider pion-nucleon scattering,  N   N. There are six elastic processes:
(a)    p     p
(c)    p     p
(e)  0  n   0  n
(b)  0  p   0  p
(d)    n     n
(f)    n     n
and four charge-exchange processes:
(h)  0  p     n
(j)    p   0  n
(g)    n   0  p
(i)  0  n     p
Since the pion carries I = 1, and the nucleon I = 12 , the total isospin can be 32 or
So there are just two distinct amplitudes here: 3, for I = 32 , and 1, for I = 12 .
From the Clebsch-Gordan tables we find the following decompositions:
   p : 11
1 1
2 2

 0  p : 10
1 1
2 2
 2/3
   p : 1 1
1 1
2 2
3 3
2 2
 (1/ 3)
3 1
2 2
 (1/ 3)
3
2
1 1
2 2
 12  2 / 3
   n : 11
1
2
 12  (1/ 3)
3 1
2 2
 0  n : 10
1
2
 12  2 / 3
 12  (1/ 3)
   n : 1 1
1
2
Reactions (a) and (f) are pure I =
a = f = 3
 12 
3
2
3
2
3
2
 2/3
1
2
 12
1 1
2 2
1
2
 12
 23
:
(1)
1
2
.
The others are all mixtures: for example
c =
1
3
3 +
2
3
1 ,
j = ( 2 / 3) 3 - ( 2 / 3) 1
(2)
The cross sections, then, stand in the ratio
 a :  c :  j  9|3 | 2 : |3 + 21| 2 : 2|3 - 1| 2
(3)
At a CM energy of 1232 MeV there occurs a famous and dramatic bump in
pion-nucleon scattering, first discovered by Fermi in 1951; here the pion and
nucleon join to form a short-lived “resonance” state the  . We know the 
carries I = 32 , so we expect that at this energy 3 »1, and hence
 a :  c :  j  9 :1: 2
(4)
(i)
Referring to equations (1) and (2), work out all the  N scattering amplitudes,
a through j, in terms of 1 and 3.
(ii)
Generalize equation (3) to include all 10 cross sections.
(iii)
In the same way, generalize equation (4).