Eigenenergy and Eigenstates

Exercise 1

In this exercises you will introduce you to the work flow of Composer and you will see how easily you can find eigen energies and eigen states of an arbitrary potential.
Unzip the zip file and start composer by running 'composer.exe'. Open the file 'Exercise 1.flow'. A block diagram with the following nodes appear:

  • Spatial dimension: A definition of the x-axis, as well as the number of points (n = 1024).
  • Potential: A definition of the potential.
  • Hamiltonian: A definition of the Hamiltonian operator.
  • ¬†Spectrum: Calculating the lowest eigen energies.
  • Energy plot: A view of the wave functions and energies.

You can in principle build these charts yourself, but we have pre-made these for you so that you can spend your time on the physics.

  1. Check that the current value of the eigen energies fit with the expected value. The program uses $m =\hbar  = 1$
  2. Try a larger value / smaller value of n = 1024 in the definition of the x-axis - the precision of the calculations depends on this.
    The infinite potential well is due to 'the steep wall' boundaries difficult to solve numerically, so here is n = 1024 a good bid. For other potentials you may choose a lower value to reduce the calculation time.
  3. Change the scalar 'a'. Does the eigen energy change as expected?
  4. Which 'a', gives a ground state energy of 1?
  5. Are the excited states as expected?
  6. Replace the potential with the harmonic oscillator, e.g. $0.5 * a {}^\wedge 2 * x {}^\wedge 2 $ (you may have to adjust the x axis). Check that the eigen energies fits as expected?
  7. Try other potentials, eg. $0.5 * a {}^\wedge 2 * abs (x) {}^\wedge q$, where q may be e.g. 1, 5, 10. Adjust 'a' so that the ground state energy becomes 1. How does the excited states distribution change for small and large values of q?