Homework Set # 3 - Due September 19

  
   
1) (a) A particle is equally likely to lie anywhere on the circumference of a circle. Consider as the z axis any straight line in the plane of the circle and passing through its center. Denote by θ the angle between the z axis and the straight line connecting the center of the circle to the particle. What is the probability that this angle lies between θ and θ+dθ?

(b) A particle is equally likely to lie anywhere on the surface of a sphere. Consider any line through the center of this sphere as the z axis. Denote by θ the angle between the z axis and the straight line connecting the center of the sphere to the particle. What is the probability that this angle lies between θ and θ+dθ?

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) Consider an isolated system consisting of a large number N of very weakly interacting localized particles of spin ½. Each particle has a magnetic moment μ which can point either parallel or antiparallel to an applied field H. The energy E of the system is then E=-(n₁-n₂)μH, where n₁ is the number of spins aligned parallel to H and n₂ the number of spins aligned antiparallel to H.

(a) Consider the energy range between E and E+δE where δE is very small compared to E but is microscopically large so that δE>> μH. What is the total number of states Ω(E) lying in this energy range?

(b) Write down an expression for lnΩ(E) as a function of E. Simplify this expression by applying Stirling's formula (lnN!≈NlnN-N).

(c) Assume that the energy E is in a region where Ω(E) is appreciable, i.e., that is not close to the extreme values +/- NμH which it can assume. In this case apply a Gaussian approximation to part (a) to obtain a simple expression for Ω(E) as a function of E.