## Delta Function Potentials p.5

### Three or more Attractive Delta Functions: Bound States

OK, two delta function "atoms", gives us a delta function molecule
"H_{2}". Does three delta function "atoms", give us
"H_{3}"? Folks whose prejudice is based on the upper
right corner of the periodic table (where: N_{2},
O_{2}, F_{2}, Cl_{2}, Br_{2}, I_{2}
live) expect no further bonding; folks who know that most
elements are metals expect atoms to continue to line up until
we've formed a small crystal: say, 10^{23} nuclei each
separated by, say, 1Å. Instead of first building the machinery
to solve the many "nuclei" problem, let's jump to the results.
We can solve for the energy eigenvalues for three delta functions
each separated by *a*. Here are the results:

On the lhs (large *a*, strong potentials, little wavefunction overlap)
we have 3 nearly degenerate energies near the isolated well energy
of -½. On the rhs
(small *a*, weak potentials, large wavefunction overlap)
we see those levels separate. for *a*=3/2 we loose one,
for *a*=1/2 we loose the other. For zero *a* we have
a unified well with a single bound state: *E*=-9/2

Now add a fourth "nucleus":

Now double that to 8 "nuclei":

We now focus on just the energy levels for *a*=4 and vary the
number of atoms: *N*=8,16,32:

Notice that as we add more atoms, we do not affect the range
of allowed energy; instead we just produce more states within the
same *energy band*. If we had *N* nuclei, we'd have
*N* states in the band. Notice that within this energy band
the *density of states* is not uniform: there are more
states per energy bin at the bottom and top of the band than in the middle.

### Wavefunctions

What do these "molecules" look like?
The most important point is that all of the wavefunctions are
spread over all of the atoms. There is no localized chemical bond
between just two atoms--every wavefunction samples the entire molecule.
Approximately speaking
the wavefunctions look like the symmetric sum of the solitary wavefunctions
amplitude modulated (i.e., multiplied) by harmonics of a sine wave that
fits the molecule. Here are the wavefunctions for *N*=8, with the
ground state presented first. (The red dots are the delta functions.)

### Binding Energy

Consider the case of assembling a molecule of *N* nuclei
and *N* electrons. If *N* is odd, one electron
must be unpaired, but generally we'll pair electrons in the lowest
available state, thus half-filling the band. Since the lower half
of the band is below *E*=-½, an average electron will have
an energy of less than -½, i.e., less than in a free atom.
Since the band energies remain fixed as more states go into the band,
the average binding energy of an electron soon reaches a constant
value. Below we plot these results for the case *a*=4.

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