Problem 1:

(a) A system is known to be in one of three states. Its probability of being in the first is 1/2 and of being in the second is 1/3. What is the density matrix for the system?

(b) Which, if any, of the following density matrices describe pure states?

Problem 2: The state of a certain system is described by the density matrix

(a) x ? , y = ?

(b) Verify whether or not the system is in a pure state.

Problem 3

(a) A system is known to be in one of two states. Show that the most general possible density matrix has the form

(b) Show that β = ±√(α(1 - α) if the system is in a pure state.

(c) Show that α = 1/2, β = 0 for an unpolarized beam of particles of spin one-half.

(d) Find α and β (in a representation in which σ_z is diagonal) for a partially polarized beam of particles such that spin parallel to the z-axis occurs twice as often as spin anti-parallel to the z-axis.

Problem 4 The state of polarization of a beam of spin one-half particles is described by the polarization vector P defined in terms of the density matrix p as follows:

P = (σ) =Tr (ρσ).

Supposing P to be determined experimentally, show that

ρ = 1/2 (1 + P.σ).