Exercise: The Finite Square Well

The applet shows the potential energy for an electron confined to a finite square well of width 0.200 nm and height 100 eV (these values appear on the Math tab, along with the electron mass = 511 keV/c²). The listing to the right of the graph includes placeholders for three stationary states for the electron in this well. In this exercise we will find the two lowest-lying bound states and determine the total number of bound states this well can support. The exercise illustrates energy quantization, i.e., only certain energies are permitted for the electron, corresponding to states which satisfy the acceptability criteria for quantum wavefunctions.

1. Show the first waveform in the list (labeled ψ₀) by right-clicking its placeholder, clicking on the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the OK button. The displayed wavefunction has a noticeable discontinuity. Since quantum wavefunctions must be everywhere continuous, the energy of this state cannot be one of the allowed energies for the electron.
2. Adjust the energy of this state upward from zero to eliminate the discontinuity. Select the Math tab, where the energy of this state is recorded as E₀ = 0. Right-click anywhere in the value field for E₀ and choose "Edit Parameter..." from the popup menu to bring up the Energy Editor. Return to the Graphics: [x] tab and re-position the editor so as to afford an unobstructed view of the waveform. Now use the slider to manipulate the highlighted digits in the energy field (labeled E) while observing the waveform. Your goal is to reduce the discontinuity to an imperceptible level. 'Fine tuning' is accomplished by adjusting the number of highlighted digits using the arrows to the immediate right of the energy field. The actual wave mismatch, expressed as a fraction of the wave value, is recorded in the editor tolerance field, labeled δψ. When no discontinuity in the waveform is evident, the energy is 'allowed' and the wavefunction is one of the bound states for the electron in this potential well. The reset button () at the lower right changes the point of discontinuity, and should be used when nearing a correct energy – see Technical Notes below. Count the number of nodes for the wavefunction to see which bound state you have found. Finish by selecting OK to end the edit session with the current settings.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor energy field, then pressing the enter key. The sum of leading and trailing digits is limited to 9.
   
   
3. Repeat the above procedure for the second placeholder in the list, labeled ψ₁, adjusting the corresponding energy E₁ until no discontinuity is evident. If you found the ground state (no nodes) in the previous step, search now for an acceptable wavefunction with the next-lowest energy; this is the first-excited state, recognized by having exactly one node. Note the different symmetries for the two states: the ground state is symmetric about the midpoint of the potential well, the first-excited state is antisymmetric about this point.
4. Repeat the procedure once more for the third placeholder, labeled ψ₂, with associated energy E₂. This time search for an acceptable wavefunction with the highest possible energy, but still localized in the well. Count the number of nodes to discover which excited state you have found. Adding one to this number (for the ground state) gives the total number of electron bound states this particular well can support.

Final form of applet...

Technical Notes

• The Numerov method is used to contruct stationary waves for a given (trial) energy. Using boundary conditions derived from the correct asymptotic form first at the left endpoint, and then again at the right endpoint, the Schrödinger equation is integrated inward to a preset match point. The right-side wave is then scaled to make its slope agree with that of the left-side wave at the match point. If the trial energy is allowed, the wavefunction values also will agree at the match point; otherwise, a discontinuity results. The [fractional] discontinuity is recorded in the editor tolerance field, labeled δψ. The technique is most accurate when the match point coincides with an extremum of the wavefunction. Each press of the reset button re-positions the match point to an extremum of the current waveform.