- Show that
1
√
2
((
1
0
)
⊗
(
0
1
)
−
(
0
1
)
⊗
(
1
0
))
is a basis of theV^0 component of the tensor product, by computing first
the action ofSU(2) on this vector, and then the action ofsu(2) on the
vector (i.e., compute the action ofπ′(X) on this vector, forπthe tensor
product representation, andXbasis elements ofsu(2)).- Show that
(
1
0
)
⊗
(
1
0
)
,
1
√
2
((
1
0
)
⊗
(
0
1
)
+
(
0
1
)
⊗
(
1
0
))
,
(
0
1
)
⊗
(
0
1
)
give a basis for the irreducible representationV^2 , by showing that they are
eigenvectors ofπ′(S 3 ) with the right eigenvalues (weights), and computing
the action of the raising and lowering operators forsu(2) on these vectors.Problem 2:
Prove that the algebraS∗(V∗) is isomorphic to the algebra of polynomial
functions on the vector spaceV.
B.6 Chapters 10 to
Problem 1:
Consider a quantum system describing a free particle in one spatial dimen-
sion, of sizeL(the wavefunction satisfiesψ(q,t) =ψ(q+L,t)). If the wavefunc-
tion at timet= 0 is given by
ψ(q,0) =C(
sin(
6 π
Lq)
+ cos(
4 π
Lq+φ 0))
whereCis a constant andφ 0 is an angle, find the wavefunction for allt. For
what values ofCis this a normalized wavefunction (
∫
|ψ(q,t)|^2 dq= 1)?Problem 2:
Consider a state att= 0 of the one dimensional free particle quantum system
given by a Gaussian peaked atq= 0
ψ(q,0) =√
C
πe−Cq2whereCis a real positive constant.
Show that the wavefunctionψ(q,t) fort >0 remains a Gaussian, but one
with an increasing width.
Now consider the case of an initial stateψ(q,0) with Fourier transform
peaked atk=k 0
ψ ̃(k,0) =√
C
πe−C(k−k^0 )2