Physics 745 - Group Theory
Solution Set 23
1. [20] The group SU(2) shows up in surprising places. Consider, for example, the
two-dimensional harmonic oscillators, which can be written in the form
H = ( a1 a1 + a2 a2 + 1)
where a1 and a2 are two operators
Physics 745 - Group Theory
Solution Set 27
1. [5] This problem has to do with demonstrating that according the isospin
symmetry, the three s all have the same mass.
(a) [2] Work out the effects of the isospin operators I on all three of the
states.
These
Physics 745 - Group Theory
Solution Set 28
1. [15] The group SU(3) contains the group SU(2) as a subgroup, and in more than
one way
(a) [7] Show that the generators T1, T2 and T3 form an SU(2) subgroup; that is,
show that [T1 , T2 ] = iT3 , etc. To save t
Physics 745 - Group Theory
Solution Set 29
1. [5] Using a weight diagram, or tensor methods (your
choice), work out the decomposition of the tensor product
3 3 into irreps.
T8
If you add the three weights to themselves, you get a total of
T3
nine possible
Physics 745 - Group Theory
Solution Set 30
1. [5] The mass of the can be predicted in terms of the parameters a and b from
eq. (4.36).
(a) [2] Find the formula for the mass in terms of a and b.
This is straightforward:
m = awijk wijk + bwijk wijl (T8 )l =
Physics 745 - Group Theory
Solution Set 25
1. [15] In the electric dipole approximation, the rate at which an atom decaying
from one state to another by the emission of a photon is given by
2
3
( I F ) = 4 IF rFI c 2 where
3
rFI = F r I
The absolute valu
Physics 745 - Group Theory
Solution Set 24
1. [5] The total angular momentum of an atom actually has three pieces: The
orbital angular momentum of the atom, the spin of the electron, and the spin of
the nucleus.
(a) [1] Suppose a hydrogen atom has a singl
Physics 745 - Group Theory
Solution Set 20
1. [10] Prove, using only the commutation relations, the first three identities (2.9)
from the notes, namely
T2 , Ta = 0,
[T3 , T ] = T ,
and T2 = TT + T32 T3
These are, in fact, three, two, and two identities r
Physics 745 - Group Theory
Solution Set 21
1. [10] In class (or the notes), I gave explicit instructions for how to find the
j
irreducible representations Ta( ) . To demonstrate that you understand this, write
2
2
2
2
explicitly T3( ) , T( ) , T1( ) , and
Physics 745 - Group Theory
Solution Set 22
1. [10] It is rare we will actually use the representation matrices ( j ) ( R ) , but
occasionally it is useful. We want to work out explicitly ( 2 ) ( R ( x ) ) , generated
1
by the generators
Ta = 1 a ,
2
0 1
Physics 745 - Group Theory
Solution Set 26
1. [10] Below is a list of reactions that are possible. You should know the charge
and baryon number of the proton (p), electron (e) and
neutron (n). Deduce the charge Q and baryon number B
p + p +n+ p
of all lis