and the letting "a" go to zero. Thus, the
Kronig-Penny hamiltonian, taken literally, isn't physical. You need to
broaden the δ-functions a little. Limiting the calculation to N
plane waves accomplishes this broadening.
A particularly interesting feature of this solution
is the very flat band pulled down below the continuum. Fig. 3 shows a
contour plot of the square of this wavefunction for the case of q
= 0. Evidently this wavefunction is a tightly bound s state hugging the
δ-function. We have made a tight-binding model! The relative
small amplitude of this orbital at the unit cell boundary means that
electrons in it can't tunnel very easily to near-neighbor sites. That's
why the band is flat. The fortran source of the code that generated the
contours is available here. The gnuplot input file
from which the plot was made is available here.
The wavefunction is generated by the formula
This is possible because (1) the eigenstate isn't
degenerate and (2) the eigenstate has a nonzero projection on
|α>. The off-diagonal Green's function matrix element, in
turn, is given by
This result is obtained by appropriately manipulating
the inversion equations:
Further manipulation yields
The latter demonstrates that this method finds all
the energy eigenvalues, not just some of them.
© 2007 R. B. Laughlin. The author grants
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with attribution to the author, for noncommercial purposes only. All
other rights, including commercial rights, are reserved to the author.