Exercise: Excited States of the Hydrogen Atom

The applet models stationary states for the electron in hydrogen-like atoms. The choice of 'Flex Units' here means the scales for distance (d) and energy (E) are related as E·d² = ћ²/2m. With d = a₀ = 0.529 Å (1 bohr), E becomes the Rydberg energy, 1 Ry = 13.6 eV, when m is the electron mass. In these units, the Coulomb energy of an electron in the field of Z protons is just V(r) = –2Z/r. To avoid the singularity at r = 0, the applet uses a truncated Coulomb potential – see 1st Technical Note below. [Formulas for V(r) and its associated parameters for hydrogen appear in the Equation View of the applet.] On the applet tab labeled Graphics: [r] this potential is plotted over the interval [0, 40 bohrs]. The listing to the right of the graph includes placeholders for four radial waves of the electron in hydrogen; these are initialized to an s wave, a p wave, a d wave, and an f wave.

Radial waves in the applet are indexed simply by the order in which they appear, which differs from the conventional (nlm) labeling of hydrogen-like states.

1. The 3s radial wave is designated g₀ in the applet. Show this waveform by right-clicking its placeholder in the list, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the "OK" button. Similarly, exhibit the 3p and 3d radial waves (g₁ and g₂, respectively). These three make up the third atomic shell and all possess the n = 3 shell energy, –1/3² = –0.111111 Ry. [The energy of the 3s wave in the applet differs from the expected value in the 5^th decimal place – can you guess why? What makes s waves peculiar in this regard?]
2. For each radial wave there is a most probable distance of the electron from the nucleus. 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 for the electron is that value of r that maximizes |g(r)|. Locate this distance for each radial wave in the n = 3 shell by right-clicking anywhere on the graph, selecting "Trace" from the popup menu, and surveying the results. Compare most probable distances for the 3s, 3p and 3d orbitals, arranging them in order of descending value. How do these results correlate with the classical picture of the electron in orbit around the nucleus? Do any of the most probable values coincide with the orbit radius given by the semiclassical Bohr theory of the atom?
3. Because the n = 4 shell wavefunctions are more extended than their n = 3 counterparts, to display them accurately requires a larger interval. Right-click anywhere on the graph background and select "Properties..." from the popup menu. Edit the "To" field to specify an upper limit of 60 bohrs for the display interval, then choose "OK".
4. The 4s, 4p, 4d, and 4f orbitals comprise the n = 4 shell, and all have energy –1/4² = –0.0625 Ry. Show the 4f radial wave – designated g₃ in the applet – by right-clicking its placeholder in the list, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the "OK" button.
5. Display the remaining n = 4 shell members by converting their n = 3 shell counterparts. Proceed to Equation View and click in the value field for the parameter E₀, the energy for the 3s stationary state. Position the cursor as appropriate and type in the new value E₀ = –0.0625, then type "Ctrl+Z" to register the changed value. Return to Graphics View, and note that the display is updated accordingly.
   
   
   As with the 3s orbital, the energy of the 4s wave may have to be 'fine tuned' a bit until no discontinuity in the waveform is evident. If so, proceed to Equation View and right click anywhere in the equation field for E₀ and select "Edit parameter.." from the popup menu. Use the slider to change the highlighted digits in the text field while noting the waveform discontinuity; the first digit highlighted can be moved left (right) using the up (down) arrows to the right of this field. The button at the lower right resets the matching point, and should be used when nearing a correct energy – see 2nd Technical Note below. Finish by selecting "OK" to end the edit session with the current settings.
   
   Similarly, reset the values for E₁ and E₂ to the n = 4 shell energy as described above; no fine tuning should be necessary in these cases. Return once again to Graphics View to inspect the n = 4 shell radial waves you have constructed.
6. Display the full electron "cloud" ψ(x,y,z) for the 4f radial wave g₃. Return again to Equation View, right click anywhere in the equation for ψ(x,y,z), and select "Plot function" from the popup menu. Specify a range from -40 to +40 [bohrs] and 45 data points for each of the cartesian variables x, y, and z, then select "OK". Rotate (mouse drag with right button) and zoom (mouse wheel rotation) the new display as necessary to explore the salient features of this three-dimensional waveform. Compare the appearance of the 4f wave with the 4s, 4p, and 4d waves, each of which 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.
7. The magnetic quantum number m₃ of the electron "cloud" defaults to zero, but can be adjusted to any positive value within the allowed range |m| ≤ l. Right-click anywhere in the graph and select "Display in Window" from the popup menu. This frees the graph to 'float' in full view while we make adjustments to the magnetic quantum number. Reposition the graph as desired and proceed once more to Equation View, where the magnetic quantum number of this state is recorded as m₃ = 0. Right click anywhere in the equation field and select "Edit parameter..." from the popup menu. Use the spinner control on the right to change the value of m₃ interactively while observing the waveform. For nonzero m₃ the waveforms are complex; what you see is just the real part of these complex-valued wavefunctions. Change to the more informative color-for-phase plotting style (see 3rd Technical Note below) by right-clicking on the waveform and selecting "Properties..." from the popup menu. Check "Color-4-Phase" in the Display Options field, then select "OK". Do you notice any trend with increasing values of m₃? How do these results correlate with the classical picture of the electron in orbit around the nucleus?

Technical Notes

• The truncated Coulomb potential used in the applet is Coulombic down to r = a and constant therafter. The difference is inconsequential provided a is small. The default (a = 0.0001 bohr) is on the order of the nuclear size, below which the Coulomb form would not be expected to be accurate anyway.
• 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 r = 0, and then again at the interval 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 parameter 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.
• 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 degree of red, green, or blue) component of an hue-saturation-brightness color model.