Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Notice that the FT is now a periodic function in the variable G, with period
G n = 2πn/a! This shouldn’t come as a surprise though: the FT of a periodic
function is a discrete function, the FT of a discrete function is periodic.
It is easy to verify that the potential in real space an be obtained back from
its FT as follows:
V (x) =
n−1 ∑
j=0
yielding the inverse FT in discretized form:
f j e iG jx , (9.20)
V j =
n−1 ∑
m=0
f m exp[2π jm n
i]. (9.21)
The two operations of Eq.(9.19) and (9.19) are called Discrete Fourier Transform,
or DFT (not to be confused with Density-Functional Theory!). One may
wonder where have all the G vectors with negative values gone: after all, we
would like to calculate f j for all j such that |G j | 2 < E c (for some suitably
chosen value if E c ), not for G j with j ranging from 0 to n − 1. The periodicity
of DFT in both real and reciprocal space however allows to refold the G j on
the “right-hand side of the box”, so to speak, to negative G j values, by making
a translation of 2πn/a.
9.2 Code: periodicwell
Let us now move to the practical solution of a “true”, even if model, potential:
the periodic potential well, known in solid-state physics since the thirties under
the name of Kronig-Penney model:
V (x) = ∑ n
v(x − na), v(x) = −V 0 |x| ≤ b 2 , v(x) = 0 |x| > b 2
(9.22)
and of course a ≥ b. Such model is exactly soluble in the limit b → 0, V 0 → ∞,
V 0 b →constant.
The needed ingredients for the solution in a plane-wave basis set are almost
all already found in Sec.(4.3) and (4.4), where we have shown the numerical
solution on a plane-wave basis set of the problem of a single potential well.
Code periodicwell.f90 1 (or periodicwell.c 2 ) is in fact a trivial extension
of code pwell. Such code in fact uses a plane-wave basis set like the one
in Eq.(9.10), which means that it actually solves the periodic Kronig-Penney
model for k = 0. If we increase the size of the cell until this becomes large with
respect to the dimension of the single well, then we solve the case of the isolate
potential well.
The generalization to the periodic model only requires the introduction of
the Bloch vector k. Our base is given by Eq.(9.10). In order to choose when to
1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/periodicwell.f90
2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/periodicwell.c
72