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 on the Math tab 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 anti-bonding 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] wave.

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 anti-bonding orbital that derives from the 'atomic' ground states.
3. Now form a non-stationary wave from these two stationary states. On the Math tab of the applet you will see the entry Ξ(E) = 1, specifying that all stationary states in the input range will be added with unit amplitude. The spectral function φ(E) obtains from screening this envelope function through the spectral filter prescribed by the current hamiltonian operator, i.e., the output of φ(E) matches that of Ξ(E) if E is an allowed particle energy, but otherwise is forced to zero. To display φ(E), right-click anywhere in its formula field and select "Plot Function" from the popup menu. Since no domain has yet been set for the [independent] variable E, you are prompted to specify one with the help of the Domain Editor. Use ~1000 data points over an interval that includes both stationary state energies that you found above. This interval must span both energy eigenvalues, but should not be unnecessarily large – see 1st Technical Note below. Press OK to confirm your selections; a new tab appears, labeled Graphics: [E], showing the spectral composition of the mixed state. The graph should consist of just two lines, one each at the energy of the bonding and anti-bonding orbitals displayed previously. Finally, return to the Math tab and right-click anywhere in the formula field for ψ(x) and again select "Plot Function" from the popup menu. This action displays (now 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 entries in the list to the right of the graph, clicking the visibility icon beside the "Real" label in the Colors | Visibilities field, then choosing OK.
5. Now explore the evolution of the non-stationary state you have constructed. The spectral composition of this state dictates its behavior, as elaborated in the 2nd Technical Note below. From the Math tab, right-click anywhere in the formula field for ψ(x), then select "Animate..." from the popup menu. This brings up the Animator Editor. Switch back to the Graphics: [x] tab and re-position the editor as necessary to afford an unobstructed view of the waveform. Click the Play button at the top right of the editor to begin evolution. Complete control over the developing waveform is afforded by the remaining editor controls: Stop , Reverse , Restart , Step Ahead , and Step Back .
   
   
   Provision also exists for adjusting the refresh rate and the elapsed time between 'clock ticks'; both can be tweaked (with the 'clock' running) to achieve the desired 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 spectral filter 'passes' stationary states with a [fractional] discontinuity not exceeding 10⁻⁶, and 'blocks' all others. Specifying too few grid points or too large an interval may result in some eigenvalues not being captured by the filtering process.
• 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÷ℏ. 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 parlance, the phase is mapped to the hue (amount of red, green, and blue) component of an hue-saturation-brightness color model.