Thermal Physics II Homework

• Homework #1 (due Tuesday, Jan. 14) Read and bookmark the course syllabus at http://euler.phys.cmu.edu/widom/teaching/33-342/. Locate the links to homework and daily class coverage. Familiarize yourself with the quantum mechanics review.
• Homework #2 (due Wednesday, Jan. 16) Read chapter 22 Quantum Ensembles and work out these quantum mechanics questions.
• Homework #3 (due Friday, Jan. 18) Read chapter 23 Quantum Canonical Ensemble sections 1, 2 and 9. Work out these questions to get practice working with bra-ket and density matrices.
• Homework #4 (due Tuesday, Jan. 21) Read chapter 23 sections 3-5. Work out this problem to calculate the entropy of a spin-1/2 particle in a field.
• Homework #5 (due Wednesday, Jan. 22) Plot the entropy S as a function of temperature T for an ideal gas of N particles in a volume V. You can find S(E,V,N) in Swendsen chapter 7, and substitute for E(T). What happens to the entropy as T goes to absolute zero?
• Homework #6 (due Friday, Jan. 24) Read sections 23.6 and 23.7. Answer these questions about the Shannon entropy.
• Homework #7 (due Monday, Jan. 27) Read sections 23.8-11.
• Homework #8 (due Tuesday, Jan. 28) Read sections 24.1-5. Solve problem 23.6.
• Homework #9 (due Wednesday, Jan. 29, Read the quantum mechanics review section discussing the Schroedinger equation.
• Homework #10 (due Friday, Jan. 31) Read sections 24.6-8. Solve problem 23.4 (quantum harmonic oscillator).
• Homework #11 (due Monday, Feb. 3) Read On the Law of Distribution of Energy in the Normal Spectrum by Max Planck.
• Homework #12 (due Wednesday, Feb. 5, note: no HW due Tuesday, Feb. 4) Read sections 19.11 to learn Monte Carlo methods and solve problem 23.5 (Monte Carlo study of quantum harmonic oscillator). Plot time series and histograms of the state distributions Requires programming so start early! (see my example)
• Homework #13 (due Friday, Feb. 7) Read sections 25.1-5. Solve this problem on normal modes.
• Homework #14 (due Monday, Feb. 10) Solve Swendsen problem 25.1, but do it for a single mode of frequency ω rather than for the full crystal (the result is the same).
• Homework #15 (due Tuesday, Feb. 11) Read sections 25.6-8. Requires programming Write a computer program to calculate the 1-D phonon density of states. (Sample code: 1 D)
• Homework #16 (due Wednesday, Feb. 12) Requires programming Generalize the Debye dispersion relation ω=vk to arbitrary dimension d and write a computer program to plot histograms of ω values for d=1, 2, 3, sampled over the full d-dimensional Brillouin zone. (Sample code: d = 1-3)
• Homework #17 (due Friday, Feb. 14) Read sections 26.1-3. Calculate the exact 1-D DOS for photons, phonons and electrons.
• Homework #18 (due Monday, Feb. 17) Requires programming Write a computer program to compute the single particle density of states for quantum gas molecules of mass m in boxes of dimension 1, 2, and 3. Hint You can reuse your previous programs but with a different dispersion relation. (see example)
• Homework #19 (due Tuesday, Feb. 18) Read sections 26.4-8. Calculate the Helmholtz free energy, internal energy, pressure and entropy for an ideal Boltzmann gas in one dimension.
• Homework #20 (due Friday, Feb. 21) Read sections 26.9-14. Solve this problem on the Boltzmann distribution.
• Homework #21 (due Monday, Feb. 24) Read sections 26.15-17. Solve this computer-based homework problem (requires programming) to study quantum state occupation numbers (see example).
• Homework #22 (due Tuesday, Feb. 25) Read sections 27.1-4. Solve this computer-based homework problem (requires programming) to study the chemical potential of an ideal Bose gas (see example).
• Homework #23 (due Wednesday, Feb. 26) Read sections 27.5-6. Demonstrate that Bose condensation does not occur in dimensions one or two.
• Homework #24 (due Friday, Feb. 28) Solve the sample exam.
• Monday, March 16, Trial run of MWF 2:30 Zoom Meeting https://cmu.zoom.us/j/652869814
• Tuesday, March 17, Trial run of T 12:30 Zoom Meeting using same meeting-id as for MWF
• Please complete the Midsemester course evaluation and return by e-mail. I regret it will not be anonymous but please understand that I truly welcome all comments.
• Homework #25 (due Wednesday, March 18, WILL NOT BE COLLECTED!) Read sections 27.7-10. (requires programming) Solve this theoretical and computer-based homework problem to study the heat capacity of a Bose gas.
• Optional Exercise Calculate the Bose-Einstein transition temperatures of these gases
• Homework #26 (due Friday, March 20, WILL NOT BE COLLECTED!) Read sections 28.1-4 and solve Swendsen problem 28.1 (chemical potential in two-level fermion gas).
• Homework #27 (due Monday, March 23) Read sections 28.5-10. Estimate the Fermi energy for a star of one solar mass. Do this first assuming the star has the radius of the sun. Repeat for a star of 1/100 times the radius of the sun (a white dwarf) and then again for a radius of 1/100/1000 times the radius of the sun (a neutron star). In which case(s) is a relativistic treatment necessary?
• Homework #28 (due Tuesday, March 24) Finish reading Swendsen chapter 28.
• Homework #29 (due Wednesday, March 25) Requires programming Review Swendsen problem 28.1 and then reproduce the results numerically as described here (sample code)
• Homework #30 (due Friday, March 27) Read Swendsen chapter 29.1-7. Requires programming Calculate the heat capacity of a Fermi-Dirac ideal gas as described here. (sample code,
• Homework #31 (due Monday, March 30) Read Swendsen chapter 29.8-11.
• Homework #32 (due Tuesday, March 31) Read Swendsen chapter 30.1-3.
• Homework #33 (due Monday, April 6) Consider a one-dimensional binary alloy with species A and B, each of energy E_A and E_B, with nearest neighbor interactions E_AA, E_AB, E_BA and E_BB. Express the total energy for an arbitrary configuration as an Ising model hamiltonian H with exchange coupling J, magnetic field h, and constant offset energy E₀.
• Homework #34 (due Tuesday, April 7) Read Swendsen chapter 30.4. Requires programming, optional, not collected Carry out an exact numerical calculation of thermodynamic properties of the finite Ising chain. (Sample code)
• Homework #35 (due Wednesday, April 8) This problem on the transfer matrix solution to the Ising chain.
• Homework #36 (due Friday, April 10) Read Swendsen chapter 19.11. Requires programming Write a computer program to simulate the Ising chain using Metropolis Monte Carlo. Compare your results with your exact numerical calculation. (Sample codes and data)
• Homework #37 (due Monday, April 13) Reread Swendsen chapter 17.1-9 on phase transitions in the van der Waals fluid, and read chapter 30.5-10.
• Homework #38 (due Tuesday, April 14) Solve this problem on correlation functions of the Ising chain.
• Homework #39 (due Wednesday, April 15) Determine the mean field critical temperature and critical exponent β for the spin 1 2D Ising model (σ = -1, 0, +1) in zero applied field.
• Homework #40 (due Friday, April 17) Read lecture 15 on series expansions from MIT open courseware. Solve this problem on the high temperature series for the Ising chain.
• Homework #41 (due Monday, April 20) This problem on the high temperature series for the 2D square lattice Ising model.
• Homework #42 (due Wednesday, April 22) Read lecture 16 on duality from MIT open courseware. See also the video lectures.
• Homework #43 (due Friday, April 24) (1) Derive a relationship between the low temperature series coefficients of free energy (f_n) and heat capacity (c_n). Hint: derivatives with respect to T or β can be expressed in terms of derivatives with respect to the low T expansion variable x=e^βJ. (2) Examine the high T series for the 3D Ising model susceptibility that comes from the On-Line Encyclopedia of Integer Sequences . Apply the ratio test to estimate the critical temperature.
• Homework #44 (due Monday, April 27) Write a computer program to simulate the 2D square lattice Ising model using Metropolis Monte Carlo. Calculate and graph the internal energy, magnetization, heat capacity and magnetic susceptibility at temperatures ranging from T/J=1 up to T/J=4.
• Homework #45 (due Tuesday, April 28) Run your 2D Ising Monte Carlo program with J=1 and T=T_c=2.269. Collect a histogram of the energy and also a histogram of the magnetization. Does the magnetization histogram look like you expect? See how the size of the lattice influences the shape of the histogram at T=T_c. Run it again at T=3J and compare your histograms with those at T_c. Be sure to run sufficiently long that your magnetization histogram appears symmetric, and also sufficiently above T_c that it differs significantly between the two temperatures.

Exams

• Midterm #1 Wednesday, March 4 (sample exam, earlier sample exam)
• Midterm #2 Friday, April 3 (sample exam, earlier sample exam)
• Final Friday, May 8, 8:30am (sample exam, earlier sample exam)