Exercise: Spherical Well Model of a Nucleus

The applet above simulates a proton bound to an atomic nucleus. The potential energy of the proton in the nucleus is modeled as a spherical well with radius 9.00 fm (1 fm = 10^–15 m) and height 30.0 MeV. These values, along with the proton mass = 938.38 MeV/c², appear on the Math tab of the applet. On the tab labeled Graphics: [r], the nuclear potential is plotted over the interval [0, 20 fm]. The listing to the right of the graph includes placeholders for three radial waves of the nuclear proton; these are initialized to an s wave, a p wave, and a d wave.

Radial waves in the applet are indexed simply by the order in which they appear, which differs from the conventional (nlm) labeling scheme for angular momentum states in a spherically symmetric potential.

In this exercise we will find the energies of the three lowest-lying states of the proton in this well and explore the corresponding wavefunctions.

1. The proton ground state must be an s-wave (l = 0), because any l > 0 results in an effective potential that is more repulsive. Show the s-wave – designated g₀ in the applet – by right-clicking its placeholder in the list, 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. Employ an automated search for the ground state energy. Switch 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 select "Edit Parameter..." from the popup menu. This brings up the Energy Editor, where the energy of this state is displayed in the energy field, labeled E. Next, activate the automated search function by typing "1.0" in the auto-search range field (labeled E±) and pressing the Enter key. The editor responds by inspecting energies in the range E ± 1.0, looking for a waveforms with fractional wave mismatch no larger than the [default] value shown in the tolerance field, labeled δψ. If the search succeeds, the new energy is displayed in the energy field and the tolerance field is updated with the actual wave discontinuity (see 1st Technical Note below); a failed search is marked by the requested tolerance being displayed in red, to indicate that the desired wave discontinuity was not, in fact, realized. In the event of a failed search, you will have to adjust the search parameters – either shift the mid-range energy E upward or widen the search range – and repeat. The ground state is nodeless; if you have found another, continue searching at lower energies. Finish by selecting OK to end the edit session with the current settings.
   
   
   When an automated search finishes (successfully or not), the editor reverts to manual mode; as a consequence, after editing any search parameters, you must return to the auto-search range field and press Enter to initiate another search using the new values.
   
   
   Even with an automated search, it is good practice to follow-up a successful search result by resetting the match point (click on ), as this will generate an improved waveform and a more accurate value for the wave discontinuity at the match point.
   
   
3. Repeat the above instructions to find the energies and display the radial waveforms for the lowest-lying p-wave and d-wave from their respective placeholders g₁ and g₂ in the list. Compare the energies you have found with that of the next-lowest s-wave to establish which energy is the first excited level. Which is the second excited level?
   
   
   If an automated search returns the value from a previous step, narrow the auto-search range entry so as to exclude that value from the search.
   
   
4. For each radial wave there is a most probable location for the proton. Each radial wave displayed in the applet is actually the effective one-dimensional matter wave g(r) = rR(r) that directly furnishes the radial probability density as P(r) = |g(r))|². Thus, the most probable distance of the proton from the nucleus is that value of r that maximizes |g(r)|. Find this distance for the ground state by right-clicking anywhere on the graph, selecting "Trace On/Off" from the popup menu, and surveying the results.
5. Display the probability density map for each radial wave found previously, beginning with g₂. Switch to the Math tab, right-click anywhere in the equation field for ψ(x,y,z), and select "Plot Function" from the popup menu. Since no domain for the cartesian variables has yet been set, this action brings up the Domain Editor. Specify a range from -20 to +20 [fm] and 55 data points for each of the cartesian variables x, y, and z, then select OK. Although this wave is real-valued, the display style defaults to color-for-phase (see 2nd Technical Note below), whereby positive and negative values are rendered in different colors (red and cyan, respectively). The most intense coloring marks those places where the probability of finding the electron is largest. Survey the results by rotating (mouse drag with right button pressed) and zooming (mouse wheel rotation, or mouse drag with both buttons pressed) the display. Probability maps for the remaining radial waves can be displayed by editing the subscript for g(..) in the equation for ψ(x,y,z), then typing Ctrl+Z to register the changed value. Note especially the symmetry of the different orbitals.
   
   
   The keyboard affords more precise control of the viewpoint. With the Ctrl key down, use the left|right arrow keys to rotate the line-of-sight about a vertical axis, and the up|down arrow keys to rotate the line-of-sight about a horizonal axis. Zoom in with the keypress Z; use Shift+Z to zoom out. Other graphics 'hotkeys' can be found by right-clicking on the graph background and inspecting the menu choices.
   
   
6. The magnetic quantum number m of the nuclear proton defaults to zero, but can be adjusted to any value within the allowed range |m| ≤ l. Go to the Math tab, where the magnetic quantum number of this state is recorded as m = 0. Right-click anywhere in the value field for m and choose "Edit Parameter..." from the popup menu; this brings up the Magnetic Editor. Return to the Graphics: [x,y,z] tab and re-position the editor to afford an unobstructed view of the waveform. Use the editor's up|down arrow controls on the right to increase|decrease the value of m interactively while observing the display. Each distinct choice for m produces a different wavefunction but with the same energy, i.e., all except the s-wave energy levels found above are degenerate.

Technical Notes

• In automated search mode, a variant of Newton's method is used to find the zeros of a function representing the waveform discontinuity at the match point. Automated mode is activated by entering a non-zero value into the auto-search range field, labeled E±. This action generates a target value for the [fractional] waveform discontinuity (the default is 10⁻⁶) that appears in the tolerance field, labeled δψ. The search proceeds within the auto-search range centered on the current value in the energy field. If successful, the new energy replaces the current one and the tolerance field is updated with the [fractional] wave mismatch at the new value. An unsuccessful search results in the requested tolerance being displayed in red to indicate that the desired wave discontinuity was not realized.
• A complex-valued function can be rendered as a single color-for-phase plot rather than two separate plots for the real and imaginary parts. In this display style, the phase of a complex number (the function value) is represented by color which, like phase angle, repeats with a definite period. [The 'colorwheel' follows the rainbow from red to green to violet, then back to red again through magenta.] In computer jargon, the phase is mapped to the hue (the amount of red, green, or blue) component of an hue-saturation-brightness color model.