Q.1: Consider a system consisting of 8 particles, each particle with spin direction up or down. (a) What is the total number of states of this system? (b) What is the number of states in which 4 particles have spin up and 4 particles have spin down?
Q.2: A system possesses three energy levels 0, ε, and 2ε, in equilibrium with a reservoir at τ = 0.5 eV, with degeneracies g(ε₁)= g(ε₃)=1 and g(ε₂)=3, ε = 1 eV. The particles are distinguishable.
a) Find the partition function of the system.
b) What is probability that the 2ε level is occupied?
c) Find the average energy of a particle.
d) What is the average energy of 10 such particles?
e) Calculate the Helmholtz free energy.
kᐢ = 1.38 ×10⁻²³ J/K= 8.617 × 10⁻⁵ eV/K
Q.3: Consider one free particle confined to a cube of side L. Find the concentration nᵥ (in terms of the quantum concentration nဢ) for which the ground state energy is equal to the temperature τ .
Q.4: There are xN atoms of type A and (1-x)N atoms of type B on a large lattice with N lattice sites (N >> 1). The atoms are randomly distributed. The entropy can be written on the form: σ = Nf(x).
a) Use the formalism of the microcanonical ensemble to find the function f(x).
b) Find the value of x for which σ is a maximum.