Question 1:
(a)
Verify that the Quark Model wave functions for the proton
with spin projection +1/2,
ψ(p, Jz=+1/2)
= (1/3√2) ( 2 u↑ u↑ d↓
- u↑ u↓ d↑
+ permutations),
and the Δ+ with spin projection +1/2,
ψ(Δ+, Jz=+1/2)
= (1/3) ( u↑ u↑ d↓
+ u↑ u↓ d↑
+ permutations),
are orthogonal.
(b)
Using spin and isospin lowering operators, derive the Quark Model
wave function
for the Λ hyperon.
[ Hint: start with the state
Σ*+ (Jz=+3/2),
and compute the wave functions for the
orthogonal
Jz=+1/2 states Σ*0 and
Σ0, which must also be orthogonal to
Λ (Jz=+1/2) ]
Question 2:
(a)
The SU(2) quark doublet ( u, d ) transforms as
( u', d' ) = R2(θ) ( u, d ),
where the rotation matrix
R2(θ)
= exp( i σ2 θ/2 ).
Show that the antiquark doublet in the conjugate representation,
( d—,
-u—),
also transforms under R2(θ).
(b)
Meson wave functions in the Quark Model are constructed as products
( u, d )
⊗
( d—,
-u—)
to give states of isospin 0 or 1, and spin 0 or 1.
Derive spin-flavor wave functions for the following:
(i) vector-isovector
(ρ+, ρ0, ρ-)
mesons;
(ii) pseudoscalar-isovector
(π+, π0, π-)
mesons;
(iii) pseudoscalar-isoscalar η meson.
(c)
Using the derived Quark Model meson wave functions, verify that the
G-parities of the π and ρ
are -1 and +1, respectively.
[Note: recall that quarks obey anticommutation relations,
so that e.g.
u↑
d—↓
= - d—↓
u↑ ]