Description
1 Identical particles I
The group of permutations of three particles is isomorphic to the group of geometric transformations that keep an equilateral triangle invariant (rotations of θ = 2π/3 and reflections that
leave one corner of the triangle invariant).
a) Using this correspondence, find that the set of 6 unitary 2 × 2 matrices which represent
the 3! = 6 permutations. Show that the determinant of the matrices give the parity of the
permutation.
b) Consider three indistinguishable spinless particles. Write down the ket |αi of the 3-
particle state in terms of products of single particle states. Assume now that the particles
are physically located at the corners of an equilateral triangle, which rotates around axis
perpendicular to the triangle plane (z axis). Considering the statistics, write down the allowed
quantum numbers for the total angular momentum Jz.
c) Suppose we have now three indistinguishable spin 1 particles and the orbital part of
the three particle state is anti-symmetric. Write down the spin part of the state in terms of
products of single particle kets.
2 Identical particles II
a) Suppose N identical non-interacting spin 5/2 particles are subjected to the potential of a 1
dimensional harmonic oscillator. Compute the total energy of the ground state.
b) Two identical non-interacting spin 1/2 particles occupy the energy levels of a 2D harmonic oscillator,
H =
X
2
i=1
P
2
i
2m
+
1
2
mω2x
2
i
,
where Pi
is the momentum of the particles and xi their coordinates (i = 1, 2). Each particle
has a wavefunction of the form
ψn(xi
, si) = φn(xi)χ(si),
where φn(x) is the orbital part (indexed by the energy level n ∈ N) and χ(s) the spin part.
Write down the 2-particle wave functions for the ground state and first excited states and their
respective energies.
3 Jordan-Wigner transformation
Consider a one dimensional lattice of localized spins S(m), where m labels the lattice site.
The spin operators for spin s = 1/2 have mixed commutation relations, so spin operators do
not describe neither fermions nor bosons.
a) Consider the spin operator S(m) for site m with components Sx(m),Sy(m) and Sz(m).
Defining the ladder operators
am = Sx(m) − iS(m)
a
†
m = Sx(m) + iSy(m),
where a
†
mam = Sz(m) + 1
2
(assume ~ = 1), show that they follow the commutation relations:
i) Bose type for different sites
h
a
†
i
, aj
i
=
h
a
†
i
, a
†
j
i
= [ai
, aj ] = 0
for i 6= j.
ii) Fermi type for spin operators on the same site,
n
ai
, a
†
i
o
= 1,
and a
2
i = (a
†
i
)
2 = 0.
b) Consider now a spin chain with periodic boundary conditions. Show that the operators
Ci = exp
iπX
i−1
j=1
a
†
j
aj
ai
C
†
i = a
†
i
exp
−iπX
i−1
j=1
a
†
j
aj
follow Fermi statistics.
This transformation is called Wigner-Jordan transformation. Write
down the inverse transformation. Show that
C
†
i Ci = a
†
i
ai
,
and
C
†
i Ci+1 = a
†
i
ai+1.
c) Transform the Heisenberg spin exchange Hamiltonian
H = J
X
N
m=1
S(m) · S(m + 1)
into a Hamiltonian of fermions. Interpret the result.
4 Coherent states
Consider (a
†
, a) as the creation/annihilation operators for bosons in a single mode. Coherent
states are defined as the eigenstates of the annihilation operator
a|zi = z|zi,
where z is a complex number and hz|zi = 1 is normalized.
a) Writing |zi as a generic superposition of {|ni}, states in the form:
|zi =
X∞
n=0
φn|ni,
where
|ni =
(a
†
)
n
√
n!
|0i,
show that
|zi = exp
−
1
2
|z|
2
exp
za†
|0i.
b) Show equivalently that
|zi = exp
za† − z
∗
a
|0i ≡ D(z)|0i,
where D(z) is a unitary operator.
c) Compute the overlap of two coherent states, hz|z
′
i.
d) Find the probability distribution Pn of finding the state |ni occupied in |zi.
