Homework Set # 3 - Due September 15

  
   
1) Consider a particle of mass m inside a cubic box of side L.

a) Obtain the density of states w(E) considering the energy states of the particle obtained using quantum mechanics.

b) Calculate the density of states w_c(E) of the particle in the classical limit.

c) Compare w(E) with w_c(E).

2) Problem 2.10 (assume that gamma is not 1).

3) Problem 3.1

4) Problem 3.2

5) Problem 3.3

6) Consider the systems A and A' in Fig.3.3.1 (Reif) and assume that system A has N particles, system A' has N' particles and the systems can exchange particles, in addition to energy, through the immobile wall that separates them. Like the total energy, the total number of particles N+N'=N^(0) is constant. Knowing that S=klnΩ and that now Ω=Ω(E,N) and that S has to be maximum in equilibrium, find the equilibrium conditions for this system. Hint: Define dΩ/dN=-βμ where μ is the chemical potential and β is (kT)^{-1}.

7. a) Calculate the characteristic function Q(k) of the uniform probability density function given by p(x)=1/(2a) for -a<x<a and 0 otherwise.

   b) Using Q(k), as shown in class, obtain the mean <x> and the variance <x^2>-<x>^2 for the probability distribution p(x).