SCHRÖDINGER'S EQUATION IN ONE DIMENSION

Quantum systems are quantized because of boundary conditions. These boundary conditions are what give rise to discrete energy levels. We will look at several one-dimensional systems, some of which you have already dealt with in class.

Schrödinger's equation -- or any differential equation, for that matter -- can be thought of as a prescription for calculating the wavefunction throughout space. Start with a given potential energy function and a given energy. If you know the value of the wave function at x = - ∞, and its derivative with respect to x, you can use Schrödinger's equation to crank out the wavefunction throughout the rest of space. The computer program QM1D does just that for one-dimensional problems. The only problem is that, for an arbitrarily-chosen energy, the wavefunction may blow up as x → ∞. Only certain values of energy will work: the eigenvalues for the system.

FIRST PART -- MOSTLY REVIEW:

Download QM1D_2007.xls and make a copy or two of it on your P: drive. Have your lab partner do the same. Do all of your work on your copy, not on the original version. Run QM1D. You will need to enable the macros when Excel opens this file up.

  1. Select the harmonic well potential. Taking into account that the wavefunction must vanish in the direction of both positive and negative infinity, find the lowest five values for energy, E, for which you obtain a "valid" wavefunction.
    1. Count the number of "antinodes" to the wavefunction (local maxima or minima). Call this n, the quantum number. (Warning: this way of defining n probably differs from the book's.) The ground state will have a single antinode. Find and record its energy and put a printout of the wavefunction in your notebook. Measure the energy to enough digits to test how well theory (the book) and experiment (this software) agree.
    2. Repeat for the next four values of n, recording both the ns and the energies that give you valid wavefunctions. Sketch the wavefunctions in your notebook, preferably on the printout you made for the ground state.
    3. Record in your notes the relation (function) that you find between E and n.
  2. Repeat all the above for the square well.
  3. Repeat all the above the half harmonic well.
    1. Compare (in words) the eigenvalues and wavefunctions of this potential to the simple harmonic well studied in part (a). Explain any similarities, referring to the "boundary conditions" for the "half harmonic well", and the symmetry of the "simple harmonic well" of part (a).
    2. Record in your notes the relation (function) that you find between E and n.
  4. Verify that the wavefunctions for the zero potential are sine waves. Measure the 'wavelength' of these waves as a function of energy. You can change the x-axis limits as you would for any Excel spreadsheet, if you need to change the scale. Come up with a theoretical formula for λ(E), given that λ = 2 π/k, where k is a quantity which is related to the energy, and verify that your "experimental" data agrees. You will need to measure the wavelength for energies covering a range of several orders of magnitude. (Note: the computer is using units for which ħ = 1 = m .)
SECOND PART -- GOING BEYOND THE TEXTBOOK:
  1. We saw in #3 of the First Part that putting an infinite barrier at the origin, for a 1D quantum system, rules out half of the solutions that you would otherwise have for a potential function that is symmetric about the origin. What happens if we do the same for the infinite square well?
    Write down the values for E that you got for the infinite square well above. Sketch the wavefunctions for each. Determine which ones will not work for the "half" infinite square well. Now write down the expression for the energies as a function of n for this system. Now, since all we've really done was to reduce the width of the well by a factor of two, what we've really done has been to develop an expression for how En depends on the thickness of the well, a. Write down that expression, then test it (a) by looking in the book and (b) by finding the eigenvalues, E, on the spreadsheet.
  2. Repeat the steps from above for the symmetric triangular well. Does E vs n fit a power law, as it does for both harmonic and square-well potentials? If so, what is the exponent of the power law? (HINT: The potential energy is a power law of x for all three systems: U(x) = Axn, with n=1, 2, and infinity for the triangular, harmonic, and square wells respectively. The power laws for E vs n for the harmonic and square wells are linear and quadratic. Thus, you would expect the triangular system to have a power law with an exponent smaller than 1. Do you? Include uncertainty.)

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