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 on the Math tab 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 on 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. Go to the Math tab, where the energy of this state is recorded as E₀ = 0. Right-click anywhere in the value field for E₀ and choose "Edit Parameter..." from the popup menu to bring up the Energy Editor. Return to the Graphics: [x] tab and re-position the editor so as to afford an unobstructed view of the waveform. Now use the slider to manipulate the highlighted digits in the energy field (labeled E) while observing the waveform. Your goal is to reduce the discontinuity to an imperceptible level. 'Fine tuning' is accomplished by adjusting the number of highlighted digits using the arrows to the immediate right of the energy field. The actual wave mismatch, expressed as a fraction of the wave value, is recorded in the editor tolerance field, labeled δψ. When no discontinuity is evident, the energy is 'allowed' and the wavefunction is one of the stationary states for the atom in this environment. The reset button () at the lower right changes the point of discontinuity, and should be used when nearing a correct energy – see 1st Technical Note below. 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.
   
   
   Extra digits can be added before or after the decimal by typing directly in the editor energy field, then pressing the Enter key. The sum of leading and trailing digits is limited to 9.
   
   
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. 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 2nd 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 two lowest-lying stationary states found 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 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.
   
   
   To clearly distinguish the non-stationary state, hide the stationary states from view (right-click on the corresponding entry in the list to the right of the graph, click the visibility icon beside the "Real" label in the Colors | Visibilities field, then click 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 3rd 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 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. In this application, you will want to increase the elapsed time by one – if not two – decades.
   
   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 4th 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. Finally, reveal the individual stationary states and observe their behavior 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 of the graph 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 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 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 results 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 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 amount of red, green, or blue) component of an hue-saturation-brightness color model.