Physics 481
Spring 2006
Problem Set 2
Soeren Prell
1. Suppose we have a spin-1 meson that decays to two identical spin-1 mesons. What are
the allowed orbital angular momenta in the final state?
2. Explain why the decay
ρ0 → π 0 π 0
is impossible.
Note, the ρ0 meson is a spin-1 particle and the π 0 meson is a spin-0 particle.
3. The mathematical description of quantum mechanical angular momentum states and
the addition of angular momenta is not unique to angular momentum. There are other
properties of quantum mechanical systems that can be described by the same formalism.
One example is the two charge states of the nucleon (the proton and the neutron), which
can be considered to be a pair of states analogous to a pair of spin- 12 states. We say that
the nucleon carries ”isospin” I = 1/2; the third component of isospin I3 (analogous to
the third component of spin Sz ), has the eigenvalues + 12 for the proton and − 12 for the
neutron:
¯
¯1
p = ¯¯
À
1
+
,
2
2
¯
¯1
n = ¯¯
À
1
−
.
2
2
The proton and the neutron are then considered to be indistinguishable particles, but the
proton is ”isospin up” and the neutron is ”isospin down”.
Suppose we combine two nucleons. From the rules of addition of angular momenta (now
applied to isospin), we know that the combination gives a total isospin of 1 or 0:
|1 + 1i = pp
1
|1 0i = √ (pn + np)
2
|1 − 1i = nn
and
1
|0 0i = √ (pn − np).
2
Experimentally, the neutron and proton form a single bound state, the deuteron, while
there is no bound state of two protons or of two neutrons.
(a) What can you conclude about the isospin state of the deuteron?
(b) Given that the complete wave function for the deuteron is the product of an isospin
state, a spin state and an orbital angular momentum (spatial) state, identify the
combinations of I, S, and L that have the correct symmetry under the interchange
of the two spin- 12 nucleons. What are the allowed values of J for the deuteron?
Which combination of S, L, and J do you think the deuteron has? Explain your
answer.
4. Problem 5.11, page 214 in Griffith - Ground-state energy of Helium.
Due Friday, January 27, 5 pm. Scores for late problem sets will be divided by 2.
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