In this paper, we use a straightforward numerical method to solve scattering models in one-dimensional lattices based on a tight-binding band structure. We do this by using the wave packet approach to scattering, which presents a more intuitive physical picture than the traditional plane wave approach. Moreover, a general matrix diagonalization method that is easily accessible to undergraduate students taking a first course in quantum mechanics is used. Beginning with a brief review of wave packet transport in the continuum limit, comparisons are made with its counterpart in a lattice. The numerical results obtained through the diagonalization method are then benchmarked against analytic results. The case of a resonant dimer is investigated in the lattice, and several resonant values of the mean wave packet momentum are identified. The transmission coefficients obtained for a plane wave incident on a step potential and rectangular barrier are compared by investigating an equivalent scenario in a lattice. Finally, we present several short simulations of the scattering process, which emphasize how a simple methodology can be used to visualize some remarkable phenomena.
REFERENCES
1.
An excellent discussion of this very issue can be found in
T.
Norsen
,
J.
Lande
, and
S. B.
McKagan
, “
How and why to think about scattering in terms of wave packets instead of plane waves
,” e-print arXiv:0808.3566 (2008
).Parts of this manuscript were published in
T.
Norsen
, “
The pilot-wave perspective on quantum scattering and tunnellng
,” Am. J. Phys.
81
, 258
–266
(2013
).2.
L. I.
Schiff
, Quantum Mechanics
, 3rd ed. (
McGraw-Hill Book Company
,
New York
, 1968
).3.
C.
Cohen-Tannoudji
,
B.
Diu
, and
F.
Laloë
, Quantum Mechanics
, 2nd ed. (English) (
John Wiley & Sons, Inc
.,
New York
, 1974
).4.
R.
Shankar
, Principles of Quantum Mechanics
, 2nd ed. (
Plenum Press
,
New York
, 1994
).5.
D. J.
Griffiths
, Introduction to Quantum Mechanics
, 2nd ed. (
Pearson/Prentice Hall
,
Upper Saddle River, NJ
, 2005
), see, in particular, problems 2.43 and 2.49.6.
S.
Gasiorowicz
, Quantum Mechanics
, 2nd ed. (
John Wiley & Sons, Inc
.,
New York
, 1996
).7.
B. H.
Bransden
and
C. J.
Joachain
, Quantum Mechanics
, 2nd ed. (
Pearson/Prentice Hall
,
Toronto
, 2000
).8.
J. S.
Townsend
, A Modern Approach to Quantum Mechanics
, 2nd ed. (
University Science Books
,
Mill Valley, CA
, 2012
).10.
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
).11.
For a variety of simulations in all areas of physics, see, for example, <https://phet.colorado.edu/en/simulations/category/new>.
12.
T. L.
Cocker
,
V.
Jelic
,
M.
Gupta
,
S. J.
Molesky
,
J. A. J.
Burgess
,
G.
De Los Reyes
,
L. V.
Titova
,
Y. Y.
Tsui
,
M. R.
Freeman
, and
F. A.
Hegmann
, “
An ultrafast terahertz scanning tunnelling microscope
,” Nat. Photonics
7
, 620
–625
(2013
).13.
F.
Marsiglio
, “
The harmonic oscillator in quantum mechanics: A third way
,” Am. J. Phys.
77
, 253
–258
(2009
).14.
An excellent description of the diagonalization process with an emphasis on visualization is provided in
Kevin
Randles
,
Daniel V.
Schroeder
, and
Bruce R.
Thomas
, “
Quantum matrix diagonalization visualized
,” Am. J. Phys.
87
, 857
–861
(2019
). In our case, we use the eig solver from matlab®. This merely requires the definition of a matrix, and returns both eigenvalues and eigenvectors.15.
Wonkee
Kim
,
L.
Covaci
, and
F.
Marsiglio
, “
Impurity scattering of wave packets on a lattice
,” Phys. Rev. B
74
, 205120
(2006
).16.
F.
Marsiglio
and
R. L.
Pavelich
, “
The tight-binding formulation of the Kronig-Penney model
,” Sci. Rep.
7
, 17223
(2017
).17.
Especially if the student has already encountered the step potential where the potential decreases, and learns that reflection still occurs, and it is impedance mismatch which gives rise to reflection. See
P. L.
Garrido
,
S.
Goldstein
,
J.
Lukkarinen
, and
R.
Tumulka
, “
Paradoxical reflection in quantum mechanics
,” Am. J. Phys.
79
, 1218
–1231
(2011
). for a more in-depth discussion of this (and several other) interesting cases.18.
In this work, we consider only Gaussian initial wave packets. More general “free” wave packets are described in
M.
Andrews
, “
The evolution of free wave packets
,” Am. J. Phys.
76
, 1102
–1107
(2008
).A delightful article describing an “Airy packet,” (which is nonspreading, but for different reasons than our Gaussian wave packet on a lattice), is by
M. V.
Berry
and
N. L.
Balazs
, “
Nonspreading wave packets
,” Am. J. Phys.
47
, 264
–267
(1979
).Another article related to this work and that is published in Ref. 15 is by
A.
Mafi
, “
Transverse Anderson localization of light: A tutorial
,” Adv. Opt. Photonics
7
, 459
–515
(2015
).19.
M.
Dowling
and
F.
Marsiglio
(2005, unpublished).20.
This is also the case in Lattice QuantumChromoDynamics.
21.
N. W.
Ashcroft
and
N. D.
Mermin
, Solid State Physics
, 1st ed. (
Brooks/Cole
,
Belmont, CA
, 1976
).22.
The caveats and subtleties associated with this statement are discussed in Ref. 16, and in connection with wave packet propagation, in Ref. 15. In a nutshell, the two extreme views of electrons in a solid are the free electron model, and the tight-binding model. Both neglect interactions between electrons in their most basic forms, but the former begins with “particles in a box,” and then tries to make everything more realistic by including effects due to the underlying periodic lattice. The latter approach begins with electrons belonging to their own atoms (hence, tightly bound) and becomes more realistic by allowing electrons to tunnel from one atom to another, thus acquiring dispersion.
23.
For simplicity we now use the nearest neighbor tight-binding model only, with no loss of generality in one dimension. Therefore,
.
24.
25.
This additional potential is above and beyond the periodic lattice implicit in the tight-binding formalism. It could be used to model a local impurity site or an interface.
26.
Second quantization is usually not covered in undergraduate quantum mechanics, but a full description is beyond the scope of this paper. It really is most useful in a many-body context, whereas here we are dealing with a single particle. The formalism is closely connected to the occupation number representation, so-called because instead of keeping track of the particle and what state it is in, one keeps track of all the states and how many particles are in each state. A proper description is available in
Philip L.
Taylor
and
Olle
Heinonen
, A Quantum Approach to Condensed Matter Physics
(
Cambridge U. P
.,
Cambridge
, 2002
).27.
D. H.
Dunlap
,
H.-L.
Wu
, and
P. W.
Phillips
, “
Absence of localization in a random-dimer model
,” Phys. Rev. Lett.
65
, 88
–91
(1990
).28.
H.-L.
Wu
and
P. W.
Phillips
, “
Polyaniline is a random-dimer model—A new transport mechanism for conducting polymers
,” Phys. Rev. Lett.
66
, 1366
–1369
(1991
).29.
H.-L.
Wu
,
W.
Goff
, and
P. W.
Phillips
, “
Insulator-metal transitions in random lattices containing symmetrical defects
,” Phys. Rev. B
45
, 1623
–1628
(1992
).30.
P. K.
Datta
,
D.
Giri
, and
K.
Kundu
, “
Nonscattered states in a random-dimer model
,” Phys. Rev. B
47
, 10727
–10737
(1993
).31.
D.
Giri
,
P. K.
Datta
, and
K.
Kundu
, “
Tuning of resonances in the generalized random trimer model
,” Phys. Rev. B
48
, 14113
–14120
(1993
).32.
W.
Kim
,
L.
Covaci
, and
F.
Marsiglio
, “
Hidden symmetries of electronic transport in a disordered one-dimensional lattice
,” Phys. Rev. B
73
, 195109
(2006
).33.
Indeed, we already have preliminary results in two dimensions, but these require considerable more computational effort.
34.
Aditi
Sharma
,
Swapna
Gora
,
Jithin
Bhagavathi
, and
O. S. K. S.
Sastri
, “
Simulation study of nuclear shell model using sine basis
,” Am. J. Phys.
88
, 576
–585
(2020
).35.
Fatih
Doğan
,
Lucian
Covaci
,
Wonkee
Kim
, and
Frank
Marsiglio
, “
Emerging non-equilibrium bound state in spin current-local spin scattering
,” Phys. Rev. B
80
, 104434
(2009
).© 2021 Author(s). Published under an exclusive license by American Association of Physics Teachers.
2021
Author(s)
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.
