6805: 1094 Activities 10

Write your name and answers on this sheet and hand it in at the end.

Work with others at your table on these activities. Argue about the answers but work efficiently!

Ising Model Simulation, part 2

Goal: More exploration with the 2-D Ising Model

  1. From the 6805 home page, go to the webpage for "2-D Ising model (js)" [not the html5 simulation]. There are various controls, among which we'll use the temperature and graph type. At the top are the current value, mean value, and variance for the energy e(t) and the magnetization per spin m(t). Two things to know: 1) to enter a precise temperature, click on the box and enter the number; and 2) when you change a parameter, press "Reset data".
  2. Play with the simulation a bit, changing the temperature (Tc = 2.27) and switching back and forth between looking at the magnetization and the energy (remember to "Reset data"). An order parameter is (usually) a quantity that is zero in one phase and non-zero in another phase. Is either e(t) or m(t) an order parameter for the Ising model phase transition?

  3. If you lived and did experiments in the Ising world at low temperature, could you tell that there was a symmetry between spin up and spin down? What about at high temperature? What does this say about symmetry in the two phases?

  4. We can calculate the energy of any particular configuration using the Ising model Hamiltonian:    E[configuration] = −J Σ⟨i,j⟩ Si Sj , where the sum is over "nearest neighbor" sites labeled by i and j, and Si is the spin (+1 or −1) on the ith site. Do you have any questions about that?.

  5. The probability of any particular energy E is given by a Boltzmann factor (as appeared in thermal distribution functions) times the number of different configurations that have that energy, Ω(E):
       P(E) ∝ Ω(E) e−E/kT   For the lowest (most negative) energy, what is Ω(E)? For what energy is Ω(E) largest?

  6. To find the most probable energy, we have to maximize P(E). We can make that easier to see by moving Ω(E) to the exponent, because e−α is maximized when α is minimized. Use z = eln z to rewrite P(E) as e−(E−TS)/kT. What is S(E)? What is the combination E− TS?

Discretization Games

Goal: Look at how to discretize functions and derivatives, and apply it to the Schroedinger equation.

  1. Note: the notebook is originally defined to show explicit results for most evaluations. As you increase the number of points used, you will want to suppress many of the outputs with ;'s to keep the notebook readable.
  2. Discretized functions. Open the Mathematica notebook discretization_games.nb and evaluate the cells in this first section.
    1. Why is DeltaX defined with Npts-1 instead of Npts?

    2. Play around with changing Npts and Xmax, re-evaluating this section each time. Formulate a rule (based on the graph) to decide if Npts is large enough.

    3. Do you have any questions about the Mathematica code in this section?

  3. Calculating first derivatives in matrix form.
    1. We'll define two types of approximate derivatives: forward derivatives and central derivatives.
      The forward derivative uses df/dx ≈ [f(x + Δx) - f(x)]/Δx while the central derivative uses df/dx ≈ [f(x + Δx) - f(x - Δx)]/2Δx
      Apply Taylor expansions to f(x + Δx) and f(x - Δx) including the second derivative to verify these results. What is the error in each case?

    2. In the notebook, set Xmax=4 and Npts=11 again, and rerun the section 1. one more time. Then step through section 2.
    3. Show (schematically) that multiplying the discrete function from section 1. by the forwardDerivative and centralDerivative matrices takes the approximate derivatives.

    4. Try a larger number of points. (Note: when you increase Npts, you need to run both sections again.) Which derivative works better? Do the errors scale as expected from your derivation above?

    5. (BONUS, only if time permits). Why do the derivative matrices fail at the endpoints? How are they fixed in the "Fix up the derivatives" subsection?

    6. Do you have any questions about the Mathematica code in this section?

  4. Calculating second derivatives in matrix form.
    1. Our approximation for the second derivative is:
      d2f/dx2 ≈ [f(x + Δx) - 2f(x) + f(x - Δx)]/(Δx)2
      Apply Taylor expansions to verify this results. What is the error?

    2. Do you understand how this works as a matrix? Why does it fail at the endpoints?

    3. Try the examples with different numbers of points. Do the errors scale as expected from your derivation above?

    4. Do you have any questions about the Mathematica code in this section?

  5. Solve the S-equation by matrix diagonalization.
    1. Suppose you have an unknown column vector ψ represented as a list of Npts values. Show (schematically) that if you multiply ψ by the matrix myHamiltonian and set it equal to energy E times ψ, then this is a discrete version of the Schroedinger equation.

    2. What will the eigenvalues and eigenfunctions of myHamiltonian correspond to?

    3. In principle there are an infinite number of energy eigenvalues for the 1-D harmonic oscillator. How many do we get here? Knowing that the exact energies in the units here are (n+1/2) with n = 0,1,2,..., which eigenvalues seem to be best determined?

    4. Make a table of the lowest three eigenvalues for Npts = 11, 51, 101, 201. How much does the accuracy improve if you (roughly) double the number of points?

    5. The plotted eigenvectors are the wave functions. Sketch them here. How do you know these are the three lowest wave functions?

    6. At the very end, the dot product of the same and different eigenvector wave functions is carried out. Try some other combinations. How do you explain the results?

    7. Do you have any questions about the Mathematica code in this section?

6805: 1094 Activities 10. Last modified: .