Homework Set # 8 - Due October 30

1. Problem 5.1

2. Consider a rotor in 2 dimensions with H=-(ћ²/2I)d²/dθ and 0≤θ≤2π.

   a) Find the eigenstates and energy levels of the system.

   b) Write the expression for the density matrix <θ'|ρ|θ> in a canonical ensemble at temperature T.

3.       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.

4. 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.

5. 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.