Introduction to Quantum Mechanics, Fall 2014
Problem Set 5
Due Thursday, November 6
Problem 1: Consider the action of SU (2) on the tensor product V 1 ⊗V 1 of two
spin one-half representations. According to the Clebsch-Gordan decomposition,
this breaks up into irreducibles as V 0 ⊕ V 2 .
1. Show that
1
1
0
0
1
√ (
−
⊗
)
⊗
0
1
1
0
2
is a basis of the V 0 component of the tensor product, by computing first
the action of SU (2) on this vector, and then the action of su(2) on the
vector (i.e. compute the action of π 0 (X) on this vector, for π the tensor
product representation, and X basis elements of su(2)).
2. Show that
1
1
0
0
1
0
0
1
1
⊗
+
⊗
),
⊗
⊗
,√ (
1
1
0
1
1
0
0
2 0
give a basis for the irreducible representation V 2 , by showing that they are
eigenvectors of π 0 (S3 ) with the right eigenvalues (weights), and computing
the action of the raising and lowering operators for su(2) on these vectors.
Problem 2: In class we described the quantum system of a free non-relativistic
particle of mass m in R3 . Using tensor products, how would you describe a
system of two identical such particles? Find the Hamiltonian and momentum
operators. Find a basis for the energy and momentum eigenstates for such a
system, first under the assumption that the particles are bosons, then under the
assumption that the particles are fermions.
Problem 3: Consider a quantum system describing a free particle in one spatial
dimension, of size L (the wavefunction satisfies ψ(q, t) = ψ(q + L, t)). If the
wave-function at time t = 0 is given by
ψ(q, 0) = C(sin(
6π
4π
q) + cos( q + φ0 ))
L
L
where C is a constant and φ0 is an angle, find the wave-function
for all t. For
R
what values of C is this a normalized wave-function ( |ψ(q, t)|2 dq = 1)?
Problem 4: Consider a state at t = 0 of the one-dimensional free particle
1
quantum system given by a Gaussian peaked at q = 0
r
C −Cq2
ψ(q, 0) =
e
π
where C is a real positive constant.
Show that the wavefunction ψ(q, t) for t > 0 remains a Gaussian, but one with
an increasing width.
Now consider the case of an initial state ψ(q, 0) with Fourier transform peaked
at k = k0
r
C −C(k−k0 )2
e
e
ψ(k, 0) =
π
What is the initial wave function ψ(q, 0)?
Show that at later times |ψ(q, t)|2 is peaked about a point that moves with
velocity ~k
m.
2