Quantum Statistics 2017, Problem set 1
Solutions to be returned to the mail box of Aleksi Vuorinen (A322) by 4pm on Tuesday, March
21st. The problems will be discussed in the exercise session of Friday, March 24th.
1. Let us review some quantum mechanics in a thermodynamic setting. Assume that we have
a gas of weakly interacting particles in thermal equilibrium at room temperature, and that
we isolate one or several of these spin 0 particles of mass m to a cubic box of size a3 .
a. Solve the energy levels of the particle from the non-relativistic Schrödinger equation.
b. Assume that the energy of the particle corresponds to the typical kinetic energy of
particles at room temperature. Estimate the number of distinct one-particle states
corresponding to this energy when m = 10−25 kg and a = 10cm.
c. Estimate the degeneracy also in a case where the box contains NA = 6×1023 particles.
2. Build the properly normalized basis function of the Hilbert space of four bosonic particles
using the single partic wave functions ψ1 (ξ) ja ψ2 (ξ), assuming that
a. the state 1 contains three bosons and the state 2 one boson,
b. both states contain two bosons.
Check the normalization of your solution.
3. Let us study the Hermitian density operator ρ̂(t). Show that
a. the corresponding state is a pure state, if ρ̂ satises ρ̂2 = ρ̂ and Tr ρ̂ = 1,
b. the expectation value of an observable Â(t) that has explicit time dependence (e.g. through coupling to an external eld), hÂi = Tr ρ̂Â, satises the equation of motion
∂ Â(t)
i
d
hÂ(t)i = h
i − h[Â(t), Ĥ]i.
dt
∂t
~
4. The Hamiltonian of spin 1/2 particles reads in suitable units
H = −4σx ,
1
σx ≡
2
0 1
1 0
,
implying that the particles are in a magnetic eld pointing in the x direction. At time
t = 0 all particles of the system are in the state sz = +1/2, described by the density matrix
1 0
ρ̂(t = 0) =
.
0 0
Determine the value of the density matrix at times t = π/8 and t = π/4.
5. Let us study a system of N massless noninteracting particles, whose single particle energies
satisfy (p) = c|p|. Determine the density of states ω(E) for this system.