Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

with V 0 > 0, b < a. The matrix elements of the Hamiltonian are given by
Eq.(4.47) for the kinetic part. by Eq.(4.48) for the potential. The latter can be
explicitly calculated:
〈b i |V (x)|b j 〉 = − 1 a
= − V 0
a
= V 0
a
∫ b/2
V 0 e −i(k i−k j )x dx (4.51)
−b/2
e −i(k i−k j )x
b/2
−i(k i − k j ) ∣
−b/2
The case k i = k j must be separately treated, yielding
V 0 (4.52)
sin (b(k i − k j )/2)
, k i ≠ k j . (4.53)
(k i − k j )/2
Ṽ (0) = V 0b
a . (4.54)
Code pwell.f90 1 (or pwell.c 2 ) generates the k i , fills the matrix H ij and diagonalizes
it. The code uses units in which ¯h 2 /2m = 1 (e.g. atomic Rydberg
units). Input data are: width (b) and depth (V 0 ) of the potential well, width
of the box (a), number of plane waves (2N + 1). On output, the code prints
the three lowest energy levels; moreover it writes to file gs-wfc.out the wave
function of the ground state.
4.4.1 Diagonalization routines
The practical solution of the secular equation, Eq.(4.32), is not done by naively
calculating the determinant and finding its roots! Various well-established,
robust and fast diagonalization algorithms are known. Typically they are based
on the reduction of the original matrix to Hessenberg or tridiagonal form via
successive transformations. All such algorithms require the entire matrix (or at
least half, exploiting hermiticity) to be available in memory at the same time,
plus some work arrays. The time spent in the diagonalization is practically
independent on the content of the matrix and it is invariably of the order of
O(N 3 ) floating-point operations for a N × N matrix, even if eigenvalues only
and not eigenvectors are desired. Matrix diagonalization used to be a major
bottleneck in computation, due to its memory and time requirements. With
modern computers, diagonalization of 1000 × 1000 matrix is done in less than
no time. Still, memory growing as N 2 and time as N 3 are still a serious obstacle
towards larger N. At the end of these lecture notes alternative approaches will
be mentioned.
The computer implementation of diagonalization algorithms is also rather
well-established. In our code we use subroutine dsyev.f 3 from the linear algebra
library LAPACK 4 . Several subroutines from the basic linear algebra library
1 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/F90/pwell.f90
2 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/C/pwell.c
3 http://www.fisica.uniud.it/%7Egiannozz/Corsi/MQ/Software/Lapack/dsyev.f
4 http://www.netlib.org/lapack/
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