A simple tight-binding model is used to illustrate how the time dependence of a state vector can be obtained from all the eigenvalues and eigenvectors of the Hamiltonian. The behavior of the eigenvalues and eigenvectors is studied for various parameters and allows us to study scatteringlike events, impurity states, and localization in disordered systems.

1.
A very good example is the article by
A.
Goldberg
,
H. M.
Schey
, and
J. L.
Schwartz
, “
Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena
,”
Am. J. Phys.
35
,
177
186
(
1967
).
2.
Leonard I.
Schiff
,
Quantum Mechanics
, 3rd ed. (
McGraw-Hill
,
Singapore
,
1968
), pp.
106
109
.
3.
Neil W.
Ashcroft
and
N. David
Mermin
,
Solid State Physics
, College ed., (
Saunders College Publishing
,
Fort Worth, TX
,
1976
), pp.
176
190
.
4.
Leonard I.
Schiff
,
Quantum Mechanics
, 3rd ed. (
McGraw-Hill
,
Singapore
,
1968
), p.
53
.
5.
Erasmo A.
de Andrada e Silva
, “
Probability current in the tight-binding model
,”
Am. J. Phys.
60
,
753
754
(
1992
).
6.
Timothy B.
Boykin
and
Gerhard
Klimeck
, “
The discretized Schrödinger equation and simple models for semiconductor quantum wells
,”
Eur. J. Phys.
25
,
503
514
(
2004
).
7.
P. W.
Anderson
, “
Localized moments and localized states
,”
Rev. Mod. Phys.
50
,
191
201
(
1978
).
8.
P. W.
Anderson
, “
Absence of diffusion in certain random lattices
,”
Phys. Rev.
109
,
1492
1505
(
1958
).
9.
Francisco A. B. F.
de Moura
and
Marcelo L.
Lyra
, “
Delocalization in the 1D Anderson model with long-range correlated disorder
,”
Phys. Rev. Lett.
81
,
3735
3738
(
1998
).
10.
Hiroaki
Yamada
, “
Localization of electronic states in a nonstationary chaotic field with long-range correlation
,”
Phys. Rev. B
69
,
014205
1
014205
8
(
2004
).
11.
Pedro
Carpena
,
Pedro
Bernaola-Galván
,
Plamen Ch.
Ivanov
, and
H. Eugene
Stanley
, “
Metal-insulator transition in chain with correlated disorder
,”
Nature (London)
418
,
955
959
(
2002
).
12.
R. G.
Endres
,
D. L.
Cox
, and
R. R. P.
Singh
, “
The quest for high-conductance DNA
,”
Rev. Mod. Phys.
76
,
195
214
(
2004
).
13.
R.
Bruinsma
,
G.
Grüner
,
M. R.
D’Orsogna
, and
J.
Rudnick
, “
Fluctuation-facilitated charge migration along DNA
,”
Phys. Rev. Lett.
85
,
4393
4396
(
2000
).
14.
Gianaurelio
Cuniberti
,
Luis
Craco
,
Danny
Porath
, and
Cees
Dekker
, “
Backbone-induced semiconducting behavior in short DNA wires
,”
Phys. Rev. Lett.
65
,
241314
1
241314
4
(
2002
).
15.
Paul
Maragakis
,
Ryan Lee
Barnett
,
Efthimios
Kaxiras
,
Marcus
Elstner
, and
Thomas
Frauenheim
, “
Electronic structure of overstretched DNA
,”
Phys. Rev. B
66
,
241104
1
241104
4
(
2002
).
16.
E.
Anderson
,
Z.
Bai
,
C.
Bischof
,
J.
Demmel
,
J.
Dongarra
,
J.
DuCroz
,
A.
Greenbaum
,
S.
Hammarling
,
A.
McKenney
, and
D.
Sorensen
,
LAPACK User’s Guide
(
SIAM Publications
,
Philadelphia
,
1999
), 3rd ed. For more information, see ⟨www.netlib.org/lapack/⟩.
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