Section 11 Exercise: The Schrödinger Equation

To extract the exercise files for this weeks exercise, do

    cd ~/ComputerPhysics
    cvs update -d

In case of problems, see the CVS update help page.

To avoid problems with "undefined procedures or functions" make sure IDL can find the necessary procedures.

Today's X-widget interface should help refresh your understanding of the basic properties of solutions to the Schrödinger equation.

Starting Up

• Start IDL (from the 11_Schroedinger directory) and type xpacket.

  NOTE: If nothing appears in the graphics window, hit QUIT and start xpacket again!

• Familiarize yourself with the GUI and the many sliders and buttons.
  • Note:You need to choose potential and packet shape (in that order) before pressing RUN.
  • The "Program caused arithmetic error: Floating underflow" that occur are normal -- not to worry!. They just reflect the fact that the tridiagonal equation system couples far away points very weakly. Also, underflows occur when computing Gaussians.
• Check the normalization before and after evolution with the NORM button (arbitrary units -- not normalized to unity).

Flat Potential: Wave Packets

The initial condition corresponds to a wave packet bouncing back and forth

• Observe how the bouncing of the wave packet creates interference at the walls
• Look at the width of the wave packet.

  
  

  Does it remain constant? Yes No Credits: 2/-1 OK

  Should it? Yes No Credits: 2/-1 OK

  Is the speed of the complex wave (the phase speed) smaller larger Credits: 2/-1 OK than the speed of the wave envelope (the group speed)?

• Measure how far the package travels, for a number of steps of size 1, twice as many steps of size 0.5, and four times as many steps of size 0.25 (choose numbers such that you can easily double/half the values).

  Is the distance the same? Yes No Credits: 2/-1 OK

  Should it be? yes no Credits: 2/-1 OK

  For a reasonably correct group speed, it is reasonable (for errors to not be noticeable in the graphics) to set the time step to less than about 1 0.1 0.01 Credits: 2/-1 OK

• In order for the complex wave to be reasonably resolved, we should also limit its wave length to be longer than about 6 grid zones.

  
  

  Does this correspond to wave packet energies of the order 0.5 1.0 2.0 Credits: 2/-1 OK (Check the code, the theory, or the plot.)

• You may want to make the plot points more visible, and "zoom in", using something similar to

     IDL> !p.psym=-1
     IDL> !x.range=[50,100]
  

  To reset back to what it was, do

     IDL> !p.psym=0
     IDL> !x.range=0
  

Gaussian Barrier

The "Gauss" button creates a potential barrier in the form of Gaussian.

• Study the reflection at, and tunneling through the Gaussian barrier. Vary the size (width) of the wave packet, within reasonable limits (it should still be a "packet" of waves).

  
  

  Does the amount of reflection / tunneling depend on the size of the packet (particle)? yes no Credits: 2/-1 OK

Harmonic Oscillator

A parabolic potential corresponds to a harmonic oscillator.

• Place the potential in the middle
• Choose a potential height at the walls well over the packet energy
• Place the initial wave packet near the center
• Let the solution run for a long time

  
  

  Does the packet spread out to an even distribution? yes no Credits: 2/-1 OK

Step Function

The step function is not infinitely steep; its shape may be controlled by the width parameter.

• Set the height of the step to about 150% of the packet energy.

  
  

  Does a steeper wall correspond to more dispersion? yes no Credits: 2/-1 OK

Double Well

• Adjust the control parameters until you have a double well, with a thin barrier in the middle. (The "potential position" should be at about 65 - 70% of the length of the box, and the width large enough to make the barrier "thin"; i.e. a few zones. The height of the barrier should be about the same as the packet energy, or slightly smaller.)
• Put the wave packet entirely within one of the wells.
• Study the leakage from one side of the double well to the other. Experiment with the height of the barrier.

  
  

  Do you believe that the solution will eventually become time independent? yes no Credits: 2/-1 OK

The following experiments should add to your understanding of the Schrödinger equation. First we are going to investigate the FFT of the signal, and then change integration algorithm to prove that the straightforward brute force does not work for this case.

• Add a common block for storing the time evolution, in your own copy of setup.pro, ala

          common TimeEvol, ff                    ; common block
          ...
          ff=complexarr(n,nt)                    ; array for (pos,time)
          ...
          ...
          ff(*,t) = phi                          ; save solution
  

• Choose a simple potential (narrow parabola or well), with the lowest possible energy, and high walls.
• Choose enough timesteps for the for wave to bounce a number of times (choose delta-t = max). QUIT and look at the Fourier transform of the solution:

          IDL> common TimeEvol, ff
          IDL> n=n_elements(ff(0,*))                   ; number of t's
          IDL> ft=ff                                   ; allocate space
          IDL> for i=0,n-1 do ft(*,i)=fft(ff(*,i),-1)  ; FFT
          IDL> shade_surf,ft*conj(ft)                  ; Shaded surface plot
          IDL> tvscl,ft*conj(ft)                       ; Flat image of the array
  

  
  

  What does this image show? The time evolution of frequencies amplitudes both amplitudes and frequencies the propagation of the wave Credits: 2/-1 OK

Now, lets change to the time integration algorithm. The previous part of the exercise showed that the implicit approach worked - preserved the total norm of the probability in time. We also hinted that this is not the case using an explicit method. Now is the time for proving that:

• Replace the time evolution by a first order (Euler) method (Eq 116 in the Notes)

          dphidt = ...
          phi = phi + dt*dphidt
  

• Try different values of dt.
• Replace the time evolution with a 4th order Runge-Kutta method (copy from Exercise 10). Include the new dfdt roiutine in the setup.pro

   
           FUNCTION dfdt,t,f,v
    
  
            ; set end points to complex(0,0.)
            ...
          END
          ...
          rk4,....
          ...
  

• Try different values of dt

OK, I have updated and saved the setup.pro procedure to use the Runge-Kutta method for time stepping Locate and upload your setup.pro file: Credits: 5/-2 max: 5

The second Project is introduced one Wednesday. You should read (have read :-) the sections on Partial Differential Equations (in Numerical Recipes -- cf. the Chapter 11 lecture notes for details).