Exercise: Inversion of the Ammonia Molecule

The applet shows the double-oscillator potential with parameters chosen to model the nitrogen atom in the ammonia molecule. [These values appear in the Equation View of the applet.] In addition, the listing to the right of the graph includes placeholders for two stationary states of the atom in this environment. In this exercise we will find the two lowest stationary states of the atom, then combine them to form a non-stationary wave that describes the nitrogen atom shuttling back and forth between its two equilibrium positions in the ammonia molecule.

1. Show the first waveform in the list (labeled ψ₀) by right-clicking its placeholder, clicking 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 atom.
2. Adjust the energy of this state upward from zero to eliminate the discontinuity. 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 energy. Reposition the graph as desired and proceed to Equation View, where the energy of this state is recorded as E₀ = 0. Right click anywhere in the equation field and select "Edit parameter.." from the popup menu. Use the slider to change the highlighted digits in the text field; the first digit highlighted can be moved left (right) using the up (down) the arrows to the right of this field. [Extra digits can be added before or after the decimal by typing directly in the text field; this example requires at least four trailing digits.] The button at the lower right resets the matching point, and should be used when nearing a correct energy – see 1st Technical Note below. When no discontinuity is evident, the energy is "allowed" and the wavefunction is one of the stationary states for the atom in this environment. Count the number of nodes for the wavefunction to see which stationary state you have found. We seek the ground state (nodeless) or the first excited state (one node). If you have found another, continue searching at lower energies. Finish by selecting "OK" to end the edit session with the current settings. Finally, restore the graph to its rightful place by clicking the close button in the upper right corner of the graph window.
3. Repeat the above procedure for the second placeholder in the list (labeled ψ₁), using it to find the other stationary state (first excited state, or the ground state), i.e., the one not found in the preceding step.
   
   
   The two stationary states are very close in energy, so it is easy to overlook one in favor of a higher excited state.
   
   
4. 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 at the energy of the two lowest-lying stationary states found 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 (you may want to hide the stationary states to clearly distinguish the new entry). Notice that this new wave is confined to the left-side oscillator well, signifying that the nitrogen atom described by this waveform is initially localized to one side of the basal plane in the ammonia molecule.
5. Inspect the evolution of the non-stationary wave you have constructed. Waveform evolution is controlled by the VCR-like button group beneath the graph: Stop , Reverse , Restart , Step Ahead , Step Back , and Play . Click on the Play button. Note how many 'ticks' of the clock elapse before the atom moves completely over to the right-side well, and how many more 'ticks' pass before the atom again takes up its original position on the left. The color variations signal a changing phase for this complex-valued wavefunction – see 2nd Technical Note below. Since each 'tick' corresponds to 1 fs = 10⁻¹⁵ s, you can readily compute the period and frequency for the 'flip-flop' of the nitrogen atom in ammonia.
6. Also observe the behavior of the individual stationary states with the passage of time. What you see is just the real part of these complex-valued wavefunctions. Change to a more informative plotting style by right clicking each stationary wave entry in the list to the right and checking "Color-4-Phase" in the Display Options field, then select "OK". Now can you explain their behavior?

Final form of applet...

Technical Notes

• 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 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 [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.
• 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 a 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.