Exercise: A Homonuclear Diatomic Molecule

The applet uses the divided square well potential to model electron states in a diatomic molecule. U = 100 eV is the height of the dividing barrier and 2w = 0.50 Å is its width. The barrier is centered at x = b and 2a = 4.50 Å is the combined width of wells plus barrier. The default case b = 0 describes potential wells of equal width (= 2.00 Å); this is the homonuclear molecule, i.e., one formed from identical atoms. All parameter values appear in the Equation View of the applet, along with the electron mass = 511 keV/c². The list to the right of the graph includes placeholders for the bonding and antibonding orbitals of the electron that derive from the 'atomic' ground states. In this exercise we will combine these orbitals to construct an initial state that describes the electron confined to one 'atom', and explore the subsequent evolution of this [non-stationary] waveform.

1. Show the first of the stationary state waveforms (labeled ψ₀) by right-clicking the topmost placeholder in the list, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the "OK" button. The displayed wavefunction has even symmetry and is nodeless; this, the 'molecular' ground state, is also the lowest lying bonding orbital.
2. Repeat the above procedure for the next placeholder in the list (labeled ψ₁). This state has odd symmetry and possesses one node; it is the antibonding orbital that derives from the 'atomic' ground states.
3. Now form a non-stationary wave from these two stationary states. In the Equation View of the applet you will see the entry φ(E) = 1, specifying that all stationary states in the input range will be added with unit amplitude. The function ψ(E) obtains from screening this envelope function through the spectral filter prescribed by the current hamiltonian operator, i.e., the output of ψ(E) matches that of φ(E) if E is an allowed particle energy, but otherwise is forced to zero. Right-click anywhere in the formula for ψ(E) and select "Plot Function" from the drop-down menu. Specify ~1000 data points over an interval that includes both energies of the stationary states you found above. [Don't make the interval unnecessarily large – just enough to contain both energy eigenvalues.] On closing the dialog, the spectral composition of the mixed state is shown in Graphics View on a separate tab, labeled Graphics: [E]. The graph for this case consists of just two lines, one each for the bonding and anti-bonding orbitals displayed previously. Finally, return to Equation View and right-click anywhere in the formula for ψ~(x) and again select "Plot Function" from the drop-down menu. This action displays (on the Graphics: [x] tab) the Schrödinger wavefunction that results from this mixture of stationary waves. Notice that the wave is confined to the left-side well, signifying that the electron described by this waveform is initially localized to one 'atom'.
4. To keep the display uncluttered, hide both stationary states from view by right-clicking their placeholders in the list, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing the "OK" button.
5. Start the 'clock' at the bottom of the applet by clicking the Play button beneath the graph. Complete control over the developing waveform is afforded by the remaining buttons on the VCR-like panel: Stop , Reverse , Restart , Step Ahead , and Step Back . [Also, right-clicking on the VCR-like panel allows for adjusting the frame rate, the number of 'clock ticks' between screen redraws, and the elapsed time between 'ticks'; all affect the speed of animation and can be tweaked (the first two with the 'clock' running) to achieve the most pleasing visual effect.]
6. Note how many 'ticks' of the clock elapse before the electron escapes completely to the adjacent 'atom', and how many more 'ticks' pass before it returns to its original position. The color variations signal a changing phase for this complex-valued wavefunction – see 3rd Technical Note below. Since each 'tick' corresponds to 1 fs = 10⁻¹⁵ s, you can readily compute the period or frequency of electron transfer between the 'atoms' of this 'molecule'.

Final form of applet...

Technical Notes

• The Numerov method is used to construct stationary waves for the 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 time dependence of any non-stationary wave is readily found if its stationary state components are known, as is the case here. At time t, a stationary state with energy E acquires an additional phase = −iEt/ћ. Each such phase-adjusted state replaces the original in the mixture to give the non-statationary wave at the new time.
• 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.