Homework Set # 3 - Due September 18

  
   
1) Problem 2.3

Additional questions for problem 2.3:

i) If the cells in phase space for the classical rotator are chosen so that there are 10 cells along the generalized coordinate what has to be the length of the unit cell along the generalized momentum axis if the total volume of the cell has to be h_0=h? Make a plot.

ii) If the angular momentum of the rotator is 20ћ so that its energy is fixed, and ΔL_z=10ћ . How many microstates fit in the allowed region using the cells of part (i)? And how many using the cells for the quantum rotator? Make a plot showing the allowed region in phase space and the cells inside for both cases.

2) Consider a system consisting of two weakly interacting particles, each of mass m and free to move in 1 dimension. Denote the respective position coordinates of the two particles by x₁ and x₂, their respective momenta by p₁ and p₂. The particles are confined within a box with end walls located at x=0 and x=L. The total energy of the system is known to lie between E and E+δE. Since it is difficult to draw a 4-dimensional phase space, draw separately the part of the phase space involving x₁ and x₂, and that involving p₁ and p₂. Indicate on these diagrams the regions of phase space accessible to the system.

3) Consider an ensemble of classical one-dimensional harmonic oscillators.

(a) Let the displacement x of an oscillator as a function of time t be given by x=Acos(ωt+φ). Assume that the phase angle φ is equally likely to assume any value in its range 0<φ<2π. The probability w(φ)dφ that φ lies in the range between φ and φ+dφ is then simply w(φ)dφ=dφ/(2π). For any fixed time t, find the probability P(x)dx that x lies between x and x+dx by summing w(φ)dφ over all angles φ for which x lies in this range. Express P(x) in terms of A and x.

(b) Consider the classical phase space for such an ensemble of oscillators, their energy being known to lie in the small range between E and E+δE. Calculate P(x)dx by taking the ratio of that volume of phase space lying in this energy range and in the range between x and x+dx to the total volume of phase space lying in the energy range E and E+δE. Express P(x) in terms of E and x. By relating E to the amplitude A, show that the result is the same as that obtained in part (a).

4) For an ensemble of N classical harmonic oscillators calculate the entropy S as a function of the total energy E. Hints: 1) Since N>>1 the difference between surface and volume can be ignored and you can evaluate the total volume of phase space inside the surface of constant E; 2) Consider the change of variables q'_i=(mω)^½q_i, and p'_i=p_i/(mω)^½; 3) the volume of a hypersphere of radius R in a d-dimensional space is given by S_dR^d/d, where S_d=2π^d/2/(d/2-1)!; 4) Consider that the unit of phase space volume is h^N.

5) Problem 5