Homework Set # 8 - Due October 25

1.       a) Calculate an expression for the entropy S in terms of the density matrix operator using that the entropy, in terms of a probability distribution is given by S=-k<ln ρ>.

b) Calculate the entropy per particle for a system of N localized particles with spin ½ in a magnetic field B.

2 In class we studied the problem of N particles with spin ½ in a magnetic field by solving the problem for one single particle and using that Z_N=Z₁^N because the particles are non-interacting. For one single particle the Hamiltonian matrix is a 2x2 matrix expanded in the basis (|↑>₁, |↓>₁), where the index 1 labels the particle.

a)If you were working with N particles what would be the dimension of the Hamiltonian matrix?

b In what basis would you calculate the matrix?

c) Now we will explicitly focus on the case N=2:

i) Write the basis that you will use to expand the Hamiltonian matrix. How many elements should your basis have?

ii) Write the Hamiltonian matrix in the basis you found in (i).

iii) Calculate e^-βH in matrix form.

iv) Calculate the trace of the matrix you found in (iii). This is Z₂. Show that Z₂=Z₁².

v) Now write the density matrix for two particles ρ₂.

vi) Use ρ₂ to evaluate the average energy of the two particles U₂ and show that U₂=2U₁.

3.A quantum mechanical system is described by a simple Hamiltonian H, which obeys H²=1.

a) Evaluate the partition function for this system.

b) Calculate the internal energy for T→∞ and for T→0.

4. In a two-dimensional Hilbert space the density operator is given by its matrix elements:

x R

R* (1-x)

a) Show the the density matrix has the expected value for its trace.

b)Calculate the entropy as a function of x and R.

5. A system is described by a density operator ρ. In this problem, the eigenvalues of this operator are either 0 or 1. The number of particles in the system is N and the volume of the system is V.

a)How many eigenvalues of ρ are equal to 1?

b) What is the entropy of the system?

c)Assume that the particles are independent. If a single particle is described by a density operator ρ_i, how do you construct ρ from ρ_i?

d) What would be the entropy if they eigenvalues of ρ where either 0 or 1/N?