Homework Set # 6 - Due October 16



  1. Problem 3.22 (Hint: Use that the integral over x from – to + infinitity of exp(-ax^4) is proportional to (a)^{-1/4}).

2. Problem 3.31

3. Problem 3.41 (Hint: Give a qualitative answer and set-up an equation for TF in terms of TS, TG, NS, and NG, where the subscripts S, and G, indicate spins and gas and F stands for the final (equilibrium) temperature achieved by both systems.

4. Problem 3.42 (Hint: compare with Problem 5 in Hw#4.)

5. This problem will help answer questions regarding the application of the principle of equipartition of energy to a macroscopic harmonic oscillator such as a pendulum or a block attached to a spring. Consider a block of mass m=0.1kg attached to a spring with spring constant k=80N/m. The spring is attached to a wall and the block is lying on a very smooth surface.

  1. Calculate the partition function for the spring/block system knowing that the Hamiltonian of the system is H=p2/2m+kx2/2. Note: we assume that the equilibrium position of the block is at x=0. Hint: you can assume that the wall where the spring is attached is located at -∞ to simplify the calculation.

  2. Using the partition function obtained in A, calculate the energy of the system. Give your result in terms of the temperature T.

  3. Now assume that the block is in a room at T=300K (room temperature). Calculate the amplitude of the oscillations using the expression for the energy obtained in B. Provide your result in meters.

  4. Now assume that the block is oscillating with an amplitude A=0.1m. What is the energy of the block?

  5. What should be the temperature, according to the principle of equipartition of energy, for the block oscillating with A=0.1m? Provide your result in degrees Kelvin.

  6. Does the fact that the block can be oscillating with an amplitude of 0.1m in a room at 300K contradict the principle of equipartition of energy? Justify your answer.

  7. Was it reasonable to assume that the wall was at -∞ in part A? Justify your answer.