Electrons in an atom are confined to distinct, quantized energy levels. When atoms form solids, the interaction of the electrons causes their energy levels to split into multiple closely spaced levels, or bands, separated by forbidden regions called band gaps. Each band contains a number of energy levels equal to the number of atoms in the solid. This model of the origin of band structure can be reproduced by using a classical array of harmonic oscillators (masses connected by springs). In this system, each oscillator plays the role of an atom and its resonant frequencies play the roles of electronic energy levels. When coupled, a system of oscillators yields a spectrum of resonant frequencies and when the number of oscillators becomes sufficiently large, the system exhibits the formation of “resonant frequency bands,” similar in structure to the energy bands of an atomic solid. We experimentally demonstrate band formation using coupled harmonic oscillators and highlight the effects of both number of oscillators and coupling strength on the band structure. Additionally, we show that experimental results of this band formation follow a theoretical analysis of the system.

1.
S. V.
Harren
,
H. E.
Dève
, and
R. J.
Asaro
, “
Shear band formation in plane strain compression
,”
Acta Metall.
36
,
2435
2480
(
1988
).
2.
M.
Imada
,
A.
Fujimori
, and
Y.
Tokura
, “
Metal-insulator transitions
,”
Rev. Mod. Phys.
70
,
1039
1163
(
1998
).
3.
L.
Chen
,
X.
Chen
,
F.
Liu
,
H.
Chen
,
H.
Wang
,
E.
Zhao
,
Y.
Jiang
,
T.
Chan
,
C.
Wang
,
W.
Zhang
,
Y.
Wang
, and
S.
Chen
, “
Charge deformation and orbital hybridization: Intrinsic mechanisms on tunable chromaticity of Y3Al5O12:Ce3+ luminescence by doping Gd3+ for warm white LEDs
,”
Sci. Rep.
5
,
11514
(
2015
).
4.
Marissa
D'Onofrio
,
Mitchell
Crum
, and
Shawn A.
Hilbert
, “
An acoustic analog for a quantum mechanical level-splitting route to band formation
,”
Am. J. Phys.
84
,
841
847
(
2016
).
5.
M.
Bayer
,
T.
Gutbrod
,
A.
Forchel
,
T. L.
Reinecke
,
P. A.
Knipp
,
R.
Werner
, and
J. P.
Reithmaier
, “
Optical demonstration of a crystal band structure formation
,”
Phys. Rev. Lett.
83
,
5374
5377
(
1999
).
6.
Paul A.
Tipler
and
Gene
Mosca
,
Physics For Scientists and Engineers
, 6th ed. (
W. H. Freeman and Company
,
New York City, NY
,
2008
), pp.
458
486
.
7.
James S.
Walker
,
Physics
, 5th ed. (
Pearson Education, Inc.
,
London, UK
,
2017
), pp.
418
446
.
8.
Randall D.
Knight
,
Physics for Scientists and Engineers a Strategic Approach
, 3rd ed. (
Pearson Education
,
London, UK
,
2013
), pp.
254
264
.
9.
John R.
Taylor
,
Classical Mechanics
(
University Science Books
,
Herdon, VA
,
2005
), pp.
161
214
.
10.
Grant R.
Fowles
and
George L.
Cassiday
,
Analytical Mechanics
, 7th ed. (
Thomson Brooks/Cole
,
Independence, KY
,
2005
), pp.
82
143
.
11.
Stephen T.
Thornton
and
Jerry B.
Marion
,
Classical Dynamics of Particles and Systems
, 5th ed. (
Thomson Brooks/Cole
,
Independence, KY
,
2012
), pp.
99
137
.
12.
Luka
Novotny
, “
Strong coupling, energy splitting, and level crossings: A classical perspective
,”
Am. J. Phys.
78
,
1199
1202
(
2010
).
13.
Andrew B.
Lockhart
,
Alexandria
Skinner
,
William
Newman
,
Daniel B.
Steinwachs
, and
Shawn A.
Hilbert
, “
An experimental demonstration of avoided crossings with masses on springs
,”
Am. J. Phys.
86
,
526
530
(
2018
).
14.
Claude
Cohen-Tannoudji
,
Bernard
Diu
, and
Franck
Laloë
,
Quantum Mechanics
(
Wiley-Interscience
,
Hoboken, NJ
,
1977
), Vol.
2
, pp.
1156
1168
.
15.
The air pumps used to power the air track can be determined at <http//www.xpower.com/product/br-292a-3-hp-inflatable-blower/>.
16.
The ultrasonic motion sensors used to record the motion of the experimental apparatus can be determined at <https://www.pasco.com/prodCatalog/CI/CI-6742_motion-sensor-ii/index.cfm>.
17.
The simulation used to determine the initial starting positions needed to stimulate the higher resonant frequencies can be determined at <https://phet.colorado.edu/sims/normal-modes/normal-modes_en.html>.
18.
Ref. 9, p. 418.
19.
Charles
Kittel
,
Introduction to Solid State Physics
, 8th ed. (
Wiley
,
Hoboken, NJ
,
2005
), pp.
162
183
.
20.
Michael J.
O'Connell
,
Sergei M.
Bachilo
,
Chad B.
Huffman
,
Valerie C.
Moore
,
Michael S.
Strano
,
Erik H.
Haroz
,
Kristy L.
Rialon
,
Peter J.
Boul
,
William H.
Noon
,
Carter
Kittrell
,
Jianpeng
Ma
,
Robert H.
Hauge
,
R.
Bruce Weisman
, and
Richard E.
Smalley
, “
Band gap fluorescence from individual single-walled carbon nanotubes
,”
Science
297
,
593
596
(
2002
).
21.
C. S.
Alexander
,
G.
Cao
,
V.
Dobrosavljevic
,
S.
McCall
,
J. E.
Crow
,
E.
Lochner
, and
R. P.
Guertin
, “
Destruction of the Mott insulating ground state of Ca2RuO4 by a structural transition
Phys. Rev. B
60
,
R8422
R8425
(
1999
).
22.
P. W.
Anderson
, “
Localized magnetic states in metals
,”
Phys. Rev.
124
,
41
53
(
1961
).
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