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Intro to Particles and Fields - ex3
May 6, 2015
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Meson Representations in the quark model
“Three quarks for Muster Mark!
Sure he has not got much of a bark
And sure any he has it’s all beside the mark.”
James Joyce, Finnegans Wake
Strangeness
In this exercise we will discuss the different possible representations of a quarkantiquark duo.
In the late 1960’s the state of the art of elementary particle physics was such
that a slew of formerly unknown particles were created in the different accelerators across the world. One brand of particles posed a very curious problem for
particle physicists:
We know that the characteristic mean lifetime of a particle is proportionate to
its mass, but some particles, such as kaons, and hyperons were created easily
enough in accelerators (i.e. their creation energy and thus mass is fairly small),
but their decay rate was smaller then expected for their mass.
This led to postulating a new eigenvalue exists for these particles - ”strangeness”.
Assume we have three types of quarks: Up, Down and Strange.
The measure ”strangeness” counts the amount of strange quarks in a particle.
Such that a strange quark donates -1 ”strangeness”, and a strange antiquark
gives a +1 ”strangeness”.
1) Write down all the possible combinations of one quark and one antiquark.
2) What are the possible spin states of the above particles?
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3) It is customary to create a separate diagram for each overall spin group
(i.e. - group all the particles with spin S = λ1 ) make such groups, and
count how many different states each have.
4) Plot a diagram where the vertical axis is ”strangeness”, and the other axis
has is a tilted charge axis like so:
Place the different states you’ve found on this graph.
5) You have now successfully classified mesons, find the particle-antiparticle
pairs in your diagram.
Hint: q1 q2 = q2 q1
6) Search Wikipedia for ”mesons” and look at the spin 0 nonet.
Name the states you’ve classified according to the spin 0 nonet, and make
sure the particle-antiparticle hold.
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Su(2) Representations and Commutation
Reminder: If |αi is an eigenvector of S 2 (Spin operator) we have:
S 2 |αi = s(s + 1) |αi
S 2 = Sx2 + Sy2 + Sz2
[Si , Sj ] = iεijk Sk
The different S operators are hermitian, with eigenvalues ± 21 .
Find a matrix representation of Sx , Sy , Sz in the basis of eigenkets in which Sz
is diagonalized. Use only the known commutation relations and the demand
for hermitian matrices.
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Spin 3/2 operators
A system of spin 32 has 4 eigenvalues - − 32 , − 12 , 12 , 32 , so the minimum dimensionality of a representing matrix is 4.
Consider the following ladder operators:
p
J+ |l, m < li = l(l + 1) − m(m + 1) |l, m + 1i
J+ |l, li = 0
J− |l, m > −li =
p
l(l + 1) − m(m − 1) |l, m − 1i
J− |l, −li = 0
1) Find the matrix representation of J+
2) Find the matrix representation of J−
3) Consider that:
J+ = Jx + iJy
J− = Jx − iJy
Represent Jx and Jy as a combination J+ and J− .
4) Using what you found previously find the matrix representation of Jx and
Jy .
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Vector operators
~ = ~r × p~,
1) Show that the angular momentum operators, as defined by L
obey the following commutation relations [Li , Lj ] = i~εijk Lk
~ = (Vi , Vj , Vk ) is called a Vector operator if and only
2) A trio of operators V
if it obeys the following commutation relations:
[Li , Vj ] = i~εijk Vk
Where Li is the angular momentum operator as applies to Vi . (for instance
if Vi is represented as a 2-dimensional matrix over C, Li is a Pauli matrix.
Prove that ~r, p~ are vector operators.
~ B,
~ both vector operators, then:
3) Prove that given A,
h
i
~·B
~ =0
Li , A
4) Prove that for a radial potential V (~r) = V (r):
[Li , V (r)] = 0
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