A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy, periodic with the lattice, contains the first reciprocal lattice vector G1=2π/a, where a is the lattice constant. The time-independent Schrödinger equation can be written in reduced variables as a Mathieu equation for which numerically exact solutions for the band structure and wave functions are obtained. The band structure has band gaps that increase with increasing amplitude q of the cosine potential. In the extended-zone scheme, the energy gaps decrease with increasing index n of the Brillouin-zone boundary ka=nπ, where k is the crystal momentum of the electron. The wave functions of the band electron are derived for various combinations of k and q as complex combinations of the real Mathieu functions with even and odd parity, and the normalization factor is discussed. The wave functions at the bottoms and tops of the bands are found to be real or imaginary, respectively, corresponding to standing waves at these energies. Irrespective of the wave vector k within the first Brillouin zone, the electron probability density is found to be periodic with the lattice. The Fourier components of the wave functions are derived versus q, which reveal multiple reciprocal-lattice-vector components with variable amplitudes in the wave functions unless q = 0. The magnitudes of the Fourier components are found to decrease exponentially as a power of n for n3 to 45 for ka=π/2 and q = 2, and a precise fit is obtained to the data. The probability densities and probability currents obtained from the wave functions are also discussed. The probability currents are found to be zero for crystal momenta at the tops and bottoms of the energy bands, because the wave functions for these crystal momenta are standing waves. Finally, the band structure is calculated from the central equation and compared to the numerically exact band structure.

1.
R. de L.
Kronig
and
W. G.
Penney
, “
Quantum mechanics of electrons in crystal lattices
,”
Proc. Roy. Soc. London A
130
(
814
),
499
513
(
1931
).
2.
J. C.
Slater
, “
A soluble problem in energy bands
,”
Phys. Rev.
87
(
5
),
807
835
(
1952
) and cited references.
3.
N. W.
McLachlan
,
Theory and Application of Mathieu Functions
(
Oxford U. P
.,
London
,
1951
).
4.
L. A.
Pipes
, “
Matrix solution of equations of the Mathieu-Hill type
,”
J. Appl. Phys.
24
(
7
),
902
910
(
1953
).
5.
G. C.
Kokkorakis
and
J. A.
Roumeliotis
, “
Power series expansions for Mathieu functions with small arguments
,”
Math. Comp.
70
(
235
),
1221
1235
(
2000
).
6.
Y. S.
Choun
, “
The power series expansion of Mathieu Function and its integral formalism
,”
Int. J. Differ. Equations Appl.
14
(
2
),
81
99
(
2015
).
8.
T. R.
Carver
, “
Mathieu's functions and electrons in a periodic lattice
,”
Am. J. Phys.
39
,
1225
1230
(
1971
).
9.
L.
Ruby
, “
Applications of the Mathieu equation
,”
Am. J. Phys.
64
(
1
),
39
44
(
1996
).
10.
M.
Horne
,
I.
Jex
, and
A.
Zeilinger
, “
Schrödinger wave functions in strong periodic potentials with applications to atom optics
,”
Phys. Rev. A
59
(
3
),
2190
2202
(
1999
).
11.
J. C.
Gutiérrez-Vega
,
R. M.
Rodríguez-Dagnino
,
M. A.
Meneses-Nave
, and
S.
Chávez-Cerda
, “
Mathieu functions, a visual approach
,”
Am. J. Phys.
71
(
3
),
233
242
(
2003
).
12.
A. A.
Cottey
, “
Floquet's theorem and band theory in one dimension
,”
Am. J. Phys.
39
,
1235
1244
(
1971
).
13.
N. W.
Ashcroft
and
N. D.
Mermin
,
Solid State Physics
(
Brooks/Cole
,
Belmont, CA
,
1976
), p.
160
.
14.
J. R.
Hook
and
H. E.
Hall
,
Solid State Physics
, 2nd ed. (
Wiley
,
New York
,
2010
), p.
103
.
15.
C.
Kittel
,
Introduction to Solid State Physics
, 8th ed. (
Wiley
,
Hoboken, NJ
,
2008
).
16.

In version 11 of mathematica used in this paper, there is a bug in the program when calculating the derivatives MC(xa)/xa and MS(xa)/xa as given by the built-in functions MathieuCPrime [ar(q),q,πxa] and MathieuSPrime [ar(q),q,πxa], respectively. They are both a factor of 2 too small. This bug was taken into account when calculating dψ(xa)/dxa.

17.
D. J.
Griffiths
,
Introduction to Quantum Mechanics
(
Pearson
,
Uttar Pradesh
,
2015
), Chap. 1, Problem 1.17.
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.