Tutorial: Two-Center Potentials

Background

Broadly speaking, a two-center potential is one having exactly two stable equilibrium points. In the vicinity of each, a classical particle experiences an attractive [restoring] force. Globally, the classical motion is determined by the particle energy that fixes the turning points, which divide the space into [disjoint] allowed and forbidden regions. A classical particle originating in one of these allowed regions will remain there forever, bound to its attractive center and lacking sufficient energy to escape the potential barrier that separates this allowed region from the next. The quantum description is not nearly so straightforward.

A diatomic molecule presents a real-world example of the classic two-center problem in the quantum domain; here two atomic nuclei compete for the same electron via Coulomb attraction. The task is to find the stationary states of an electron of the molecule in terms of the atomic wavefunctions, i.e., the stationary states of an electron in a single, isolated atom. To illuminate the key ideas and basic principles, we will forego the details of the Coulomb attraction, replacing it here with the potential of a simple square well. We examine first the stationary states of an electron bound by this square well, then explore how these states are modified when a second square well is present alongside the first.

Isolated Square Well

Restricting the space available to a particle typically leads to standing wave patterns for its associated matter wave. These standing waves can exist only for certain frequencies, with the consequence that particle energy becomes quantized. Elementary discussions of energy quantization invariably begin with a treatment of the infinite square well. The more realistic finite square well tends to be passed over because of its increased mathematical demands. The applet below allows us to study the properties of the finite well without the considerable mathematical overhead that would be required by an analytical treatment.

No Java support detected The finite square well potential V(x) is shown on the Graphics: [x] tab. The well depth U = 100 eV and width L = 2.00 Å appear on the Math tab of the applet, along with the particle mass = 511 keV/c2 which we recognize as the value for an electron. The list to the right of the graph has placeholders for three stationary states of the electron in this well.

To show a stationary state ψn, right-click on its placeholder, click on the visibility icon beside the "Real" label in the Colors | Visibilities field, then choose the OK button. The waveform with no nodes (zeros) is the stationary state with the least energy; this is the ground state for the electron in this well. The waveform with one node is the first excited state for the electron; it has the next lowest energy. The second excited state has the next lowest energy and a waveform with two nodes, etc. This is a general rule of energy quantization in one dimension: the stationary states can be arranged in order of increasing energy according to the number of nodes exhibited by their wavefunctions.

The actual energies of these states (in eV) appear on the Math tab as E0, E1, and E2, respectively. These energies are among the special values leading to waveforms that satisfy the acceptability criteria for quantum wavefunctions (boundedness, continuity). Try changing the energy to see what effect this has on the waveform. Go to the Math tab, right-click anywhere in the value field for E0 and choose "Edit Parameter..." from the popup menu to open the Energy Editor. Return to the Graphics: [x] tab and re-position the editor to afford an unobstructed view of the waveform. Now use the slider to manipulate the highlighted digits in the editor energy field (labeled E) while observing the waveform. Notice how even small deviations from the true energy introduce recognizable discontinuities in the associated waveform – the actual wave mismatch, expressed as a fraction of the wave value, is recorded in the editor tolerance field, labeled δψ. Select the Cancel button to close the editor and restore the original energy for this state.

The middle of the square well is a symmetry point for the potential V(x): the left side of the well can be recovered from the right side by reflection through the midpoint. The midpoint is also a symmetry point for the stationary states. Like V(x), the ground state ψ0(x) and second excited state ψ2(x) are symmetric or even about this point. The first excited state ψ1(x) is antisymmetric or odd (reflection about the midpoint, followed by inversion). In fact, the symmetry alternates between even and odd as the energy increases.

One + One > 2!

Now suppose our electron sees not one but two identical square wells. How does this affect the stationary states? The double-well potential is shown in the applet below. Once again, there are placeholders in the list to the right for several stationary states of the electron. But before showing them, lets speculate on their properties. In particular, how are the double-well stationary states and energies related to those for the single square well?

If the wells are very far apart, you may think the answer is obvious. The stationary states will look much like they did before, but doubled in number. There should be two ground states, one with its waveform localized in the left side well and another with its waveform localized in the right side well. And each of these would have the same energy (essentially that of the single well ground state) – the ground state energy is doubly degenerate. Indeed, the same should be true for all the stationary states, not just the ground state. With decreasing well separation, the effects are more pronounced but not qualitatively different. The waveforms become more distorted and the energies deviate more from the single well values, but every level of the double well remains doubly degenerate, corresponding to waveforms localized in the left or right side compartment. Now lets take a look.

No Java support detected Reveal the stationary states as before (right-click the placeholders, click on , then OK). For the most part, the results are not at all what we expected! Where have we gone astray?

Closer inspection shows that our reasoning ignores the symmetry inherent in the situation. The double-well potential is a symmetric one, with a symmetry point midway between the two square wells. The ground state waveform must therefore be symmetric about this point – the electron must be equally distributed among the two compartments, in total disagreement with our earlier prediction. The key to understanding this rather strange twist of logic lies with the phenomenon of tunneling, a purely quantum notion beyond the reach of our intuition (which, after all, is rooted in classical physics). Were the electron confined to one compartment as we thought, it would not remain there indefinitely. No matter how far apart the wells, there is some nonzero probability for the electron to tunnel through the intervening barrier to take up residence in the other compartment! That the electron behaves this way is testimony to its decidedly non-classical nature. Indeed, all that which we have come to call "matter" is like this – it is the very essence of the quantum mystery. Continuing with the argument, our electron, having tunneled to the other compartment, now also has some liklihood of tunneling back to take up its original position! The electron localized to one potential well does not describe a stationary state at all, but a recurrent one. By extension, the stationary [steady] state must be one for which the electron has an equal chance of tunneling forward or back, implying for the ground state the equal distribution among compartments that we see in the applet.

Similarly, the first excited state must be antisymmetric with a single node [at the symmetry point], again implying an equal distribution of the electron among the two compartments. The central node here means the electron in the first excited state spends more time in the vicinity of the wells than between them, compared to the [nodeless] ground state; the result is a different [actually higher] electron energy. Moreover, the ground state and the first excited state both must arise from the ground state of the single square well (because our reasoning is valid even if the single well supports only one bound state). The energy difference between them [the "splitting"] ΔE derives from the interaction of the two square wells. Because of tunneling, the level degeneracy we anticipated earlier arises only for wells that are infinitely separated [no interaction]; for wells far apart the interaction is small and ΔE nearly (but not exactly) zero. In all cases, ΔE straddles the ground state energy of the single well (compare the applet energies of these two states with the ground state energy of the single well found earlier). To summarize, the ground state of the single square well gives rise to a pair of states for the double well, one with energy below and the other with energy above that of the single well ground state energy. And both these double-well states are extended, meaning that the electron described by these states can be found in either compartment with comparable probability.

The preceding arguments are not limited to the ground state. Each stationary state of the single well leads to a pair of [extended] stationary states for the double well, split in energy according to the strength of interaction between the two wells. The lowest energy member of the double-well duo is often referred to as the bonding orbital, because it is the one populated by electrons in the covalent bonding of atoms to form molecules; the member with higher energy is the anti-bonding orbital. In this context, the double well serves as an idealized model for the potential energy of electrons in a diatomic molecule formed from like atoms (homonuclear molecule). The bonding orbitals, marked by significant wave amplitude in the region between the atomic sites, provide the "glue" for the covalent bond that holds the two atoms together as a molecular unit.

Non-equivalent Centers

What becomes of the stationary states when the two wells are not identical? The next applet holds the answer. This is our double well potential again, but with the separator moved off of center (the height and width are unchanged at 100 eV and 0.50 Å, respectively). The wells now have different widths, 2.25 Å and 1.75 Å; in the context of chemical bonding, this model describes the potential energy of electrons in a diatomic molecule formed from two dissimilar atoms (heteronuclear molecule).

No Java support detected Reveal the stationary states as before (right-click the placeholders, click on , then OK). The symmetry of the potential is gone, and with it the symmetry of the stationary states. Still, the ground state is nodeless, the first excited state shows one node, etc. For the bonding orbital, the stronger (wider) well is more effective in competing for the electron, as indicated by the larger amplitude of the waveform there. The electron spends more time in the stronger well; the heteronuclear molecule is polar.

Recurrent States

Finally, let us return to the case of two identical wells and suppose our electron is known to be in one compartment (because we put it there!). What happens to it? The starting wavefunction for the electron must be concentrated near one of the wells. The initial state is not extended, but localized; it cannot be any one of the stationary states of the double well. The details of the initial state will naturally depend on its history (how it was prepared), but the important feature for the present discussion is its localization. To reproduce that in a simple way we add together the ground and first excited states of the double well. The contributing states are represented in the graph below by their amplitudes. Since there are only two states being added, the "plot" consists of just two lines (same height for equal amplitude) at the appropriate energies. Click the Graphics: [x] tab to see the waveform that results from this addition; as advertised, it describes an electron initially localized to just one [the left-side] compartment.

No Java support detected To explore the evolution of this localized state, go to the Math tab, right-click anywhere in the formula field for ψ(x), then select "Animate..." from the popup menu. This opens 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.

The changing colors signal a complex-valued wavefunction: the phase of a complex number (the function value) is represented here 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.

After some time, the initial wavefunction completely re-assembles in the right-side compartment – the electronic state has inverted, the electron having tunneled from the left-side compartment through the intervening barrier to arrive on the other side. This inversion is followed by another, eventually returning the system to its original state; the total elapsed time is called the recurrence time. The reciprocal of the recurrence time gives a recurrence frequency (f). For widely separated wells, the recurrence time is large and the frequency small. Larger recurrence frequencies result for wells that interact more strongly. From the stationary state viewpoint, the interaction between wells is measured by the energy splitting ΔE of the two stationary states that were combined to form this recurrent one. Thus, we might expect that f and ΔE are simply related; in fact, they are proportional, with Planck's constant (h) the constant of proportionality: ΔE = hf. See if you can verify this relationship using data supplied by the applet.

The current animator time appears on the Math tab as the value of the variable t; the individual stationary state energies also are recorded here. All numerical values are given in the system of units displayed at the bottom of the applet.