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) can be adapted to the application at hand, but always are related as E·d² = ℏ²÷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 simply V(r) = –2Z÷r. To avoid the singularity at r = 0, the applet uses a truncated Coulomb potential – see 1st Technical Note below. V(r) and its associated parameters for hydrogen appear on the Math tab of the applet. On the tab labeled Graphics: [r], this potential is plotted over the interval [0, 60 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, respectively.

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

In this exercise we will explore the excited state wavefunctions that make up the 3rd and 4th shells of hydrogen (specified by principal quantum number n = 3 and n = 4 in the conventional scheme).

1. The 3s radial wave is designated g₀ in the applet. Show this waveform 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. 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 On/Off" from the popup menu, and surveying the results. Compare most probable distances for the 3s, 3p and 3d electrons, 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 semi-classical Bohr theory of the atom?
3. 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 on the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing OK.
4. Display the remaining n = 4 shell members by converting their n = 3 shell counterparts. Switch to the Math tab 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 the Graphics: [r] tab and confirm that the display has changed. What difference(s) do you see?
   
   
   As with the 3s orbital, the energy of the 4s wave can be 'fine tuned' a bit to minimize the discontinuity (although none is visually evident). To do so, go to the Math tab, right-click anywhere in the value field for E₀ and select "Edit Parameter..." from the popup menu to activate the Energy Editor. Return to the Graphics: [r] tab and re-position the editor so as to afford an unobstructed view of the waveform. Use the slider to manipulate the highlighted digits in the energy field (labeled E). 'Fine tuning' is accomplished by adjusting the number of highlighted digits using the arrows to the immediate right of the energy field. Your goal is to minimize the wave discontinuity recorded in the editor tolerance field, labeled δψ. The reset button () at the lower right changes the point of discontinuity, 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 the Graphics: [r] tab to inspect the n = 4 shell radial waves you have constructed.
5. Display the full electron 'cloud' ψ(x,y,z) for the 4f radial wave g₃. 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 -50 to +50 [bohrs] 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 3rd Technical Note below), whereby positive and negative values are rendered in different colors (red and cyan, respectively). This is a [probability] density map, with the most intense coloring marking 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.
   
   
   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. Experiment with some advanced graphics options to explore the salient features of this three-dimensional waveform. Right-click anywhere in the graph background and select "Trace On/Off" from the popup menu. This puts the graphics in 'verbose' mode, allowing us to read the wave value at the location of the tracer [red ball], the coordinates of which are also displayed. The tracer is confined to the shaded plane, but can be moved around within that plane using the mouse. The tracer plane is perpendicular to a cartesian axis, and can be shifted along that axis using the up|down arrow keys in the panel to the right of the graph. The plane is a 'slice' through the full waveform. To see what this means, shift the plane to x = 0.0 and press Ctrl+Enter. This action 'locks' the variable x to its current value [zero], leaving a function of the two remaining variables y and z; it is that function that is now displayed – in conventional form, with function values plotted along the remaining axis. To restore the density map, 'unlock' the variable x by pressing Shift+Ctrl+Enter. Finish by pressing Ctrl+V to turn off tracing.
7. Compare the appearance of the 4f 'cloud' with the 4s, 4p, and 4d 'clouds'. To facilitate this comparison, right-click anywhere in the graph and select "Display in Window" from the popup menu; this frees the graph to 'float' in full view. Reposition the graph as desired, then return to the Math tab. Edit the subscript for g(..) in the equation field for ψ(x,y,z), then type Ctrl+Z to register the new value and update the display. Note especially the symmetry of the different orbitals. When you have finished, restore the graph to its rightful place by clicking on the close button in the upper-left corner of the 'floating' window.
   
   
   The 'floating' window can be enlarged to fill the screen by clicking on the maximize button in the upper-left corner of the window.
   
   
8. The magnetic quantum number m of the electron 'cloud' 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 so as 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. Can you identify any trends? 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 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 amount of red, green, or blue) component of an hue-saturation-brightness color model.