Exercise: Proton in an Atomic Nucleus

The applet simulates a proton confined to the nucleus of an atom. The nuclear potential is modeled here as a square well with height U = 26.0 MeV and width L = 10.0 fm (1 fm = 10⁻¹⁵ m = 10⁻⁶ nm). These values appear on the Math tab of the applet, along with the proton mass = 938.38 MeV/c². On the Graphics: [x] tab the square well potential energy V(x) is plotted over the interval [–10 fm, +20 fm]. The listing to the right of the graph includes placeholders for two stationary states of the proton in this well. In this exercise we will find the two lowest stationary states of this nucleon and confirm the notion of energy quantization for this case.

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 nuclear proton.
2. Adjust the energy of this state upward from zero to eliminate the discontinuity. Go to 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 is evident, the energy is 'allowed' and the wavefunction is one of the stationary states for the proton in this 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 stationary state you have found. We want the ground state (nodeless) or the first excited state (one node). If you have found another, continue searching at lower energies. 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, using it to find the other stationary state (first excited state, or the ground state), i.e., the one not found in the preceding step.

Final form of applet...

Technical Notes

• The Numerov method is used to construct 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.