Calculation of band structures by a discrete variable representation
based on Bloch functions
Hervé Le Rouzoa兲
Laboratoire de Photophysique Moléculaire du CNRS, Laboratoire associé à l’Université Paris-Sud,
Université Paris-Sud, bât. 210, 91405 Orsay Cedex, France
共Received 7 July 2004; accepted 11 June 2005兲
The propagation of waves in periodic media is restricted to allowed bands of energy or frequencies
separated by forbidden gaps. We propose a discrete variable representation, based on Bloch
functions, that is suitable for the calculation of band structures in one-dimensional systems. The
solutions are obtained by a single diagonalization without any integration. The method only needs
the values of the potential at grid points. Applications to the standard Krönig-Penney and Mathieu
potentials, and to the nonanalytical case of a soft Coulomb potential, show that very good accuracy
is achieved with moderate grid sizes. © 2005 American Association of Physics Teachers.
关DOI: 10.1119/1.1994858兴
I. INTRODUCTION
The propagation of various kinds of waves in periodic
materials exhibits general features such as the existence of
band gaps, that is, energy ranges in which waves cannot
propagate. These forbidden band gaps are separated by allowed bands, which together constitute the band structure of
the system. These band structures have first been recognized
for electrons in solids.1 Recently, their study has received
increasing interest for electromagnetic waves in periodic dielectric media, or photonic crystals,2 and for acoustic waves
in phononic crystals.3
In this article, we compute band structures with the aid of
a discrete variable representation 共DVR兲 based on Bloch
functions. We consider the time independent Schrödinger
equation for a one-dimensional particle of mass ␮
−
ប 2 d 2␺
+ V共x兲␺共x兲 = E ␺共x兲.
2␮ dx2
共1兲
The solution of eigenvalue problems of the form given in Eq.
共1兲 for an eigenstate ␺共x兲 with energy E is a major goal in
quantum physics and chemistry. Usually, ␺共x兲 is expanded
into a finite set of basis functions ␾n共x兲:
N
␺共x兲 = 兺 cn␾n共x兲.
共2兲
n=1
The unknown coefficients cn can be determined variationally,
for example, by either the Rayleigh-Ritz method4 if ␺ is a
bound state 共E is an unknown discrete value兲, or the variational R-matrix method5 if ␺ is a free state 共E is continuous兲.
In both cases, we first compute the matrix representation of
the Hamiltonian operator with matrix elements
冓冏
Hmn = ␾m −
冏冔
ប2 d2
+ V共x兲 ␾n = Tmn + Vmn .
2␮ dx2
Am. J. Phys. 73 共10兲, October 2005
II. THE GENERIC DISCRETE VARIABLE
REPRESENTATION
The discrete variable representation is defined by a set of
N basis functions ␾n共x兲,
␾n共x兲,
共n = 1,2,…,N兲
http://aapt.org/ajp
共4兲
together with a N-point quadrature, with real abscissae xk and
positive real weights wk:
共xk,wk兲,
共k = 1,2,…,N兲.
共5兲
The basis functions and grid points have to describe the
desired solution ␺共x兲 of Eq. 共1兲, on the interval 共a , b兲, which
is possibly infinite. The basis functions are orthonormal on
the interval 共a , b兲:
共3兲
To be able to keep only a few terms in Eq. 共2兲, the basis
set must be sophisticated, which might lead to difficulties in
evaluating the matrix elements in Eq. 共3兲. A discrete variable
representation bypasses this difficulty because the Hamiltonian matrix evaluation does not require any integration: the
kinetic energy matrix T is known analytically, and the potential is simply computed at the grid points to produce a diag962
onal matrix. The method was developed for molecular quantum mechanics by Light and co-workers in the 1980s.6
In this paper, we solve Eq. 共1兲 for an electron in a crystal
lattice, that is, for a spatially periodic potential, by constructing a discrete variable representation based on Bloch basis
functions. The Fourier basis discrete variable representation
was introduced for periodic problems.7–10 Other trigonometric basis sets also are available.11,12
In Sec. II a brief derivation of the generic discrete variable
representation is presented. In Sec. III we treat an arbitrary
periodic potential and derive explicit formulas for Bloch basis functions. In Sec. IV we apply the Bloch-discrete variable
representation to the Krönig-Penney and Mathieu potentials,
for which analytical solutions are known, as well as to the
more difficult soft Coulomb periodic potential. Our summary
and conclusion are given in Sec. V.
具兩␾m兩␾n典 =
冕
b
a
N
␾m* 共x兲␾n共x兲dx ⯝ 兺 wk␾m* 共xk兲␾n共xk兲
k=1
= ␦mn ,
共6兲
where ␾*n denotes the complex conjugate of ␾n.
Suppose that it is possible to construct from the set ␾n共x兲
new basis functions um共x兲 that satisfy13
© 2005 American Association of Physics Teachers
962
um共xk兲 =
␦mk
冑w k
共7兲
.
Each function um共x兲 is localized around the discrete grid
point xm; it is a continuous analog of the Dirac delta function
␦共x − xm兲. The matrix representation of any multiplicative op-
We can express the matrix representation of the kinetic
energy operator Tmn in Eq. 共3兲 in the new basis where V̂ is
diagonal. By definition
冓冏 冏冔
um
erator, say the potential V̂, can be shown to be diagonal:
Vmn = 具um兩V̂兩un典
共8a兲
N
N
␦mk
d2
*
u n = 兺 w ku m
共xk兲u⬙n共xk兲 = 兺 wk
u⬙n共xk兲
冑
dx2
wk
k=1
k=1
= 冑wmu⬙n共xm兲.
共14兲
We take two derivatives of Eq. 共12兲 and find
N
N
N
*
= 兺 w ku m
共xk兲V共xk兲un共xk兲 = 兺 wk
k=1
k=1
␦mk
␦nk
冑wk V共xk兲 冑wk
共8b兲
N
= 兺 ␦mkV共xk兲␦nk = V共xm兲␦mn .
共8c兲
k=1
u⬙n共x兲 = 冑wn 兺 ␾*k 共xn兲␾⬙k共x兲,
so that the matrix representation of the kinetic energy operator in the basis um is
N
Tmn = −
As a special case, the representation of the position operator
x̂ is
Xmn = 具um兩x̂兩un典 = xm␦mn .
共9兲
Equation 共9兲 often is used as the starting point for the derivation of popular discrete variable representations that rely
on orthogonal polynomials and Gaussian quadratures.14
In addition to giving a diagonal representation of V̂, Eq.
共7兲 leads to the attractive property that the potential matrix V
is given without any integration. Only a simple evaluation at
the grid points is needed, in contrast to other methods where
the calculation of potential matrix elements is a major difficulty, especially for the Coulomb potential.
Now we show how the functions um can be expressed in
terms of the original basis set ␾n. Let c̃n be the unknown
coefficients in the expansion
ប2
冑wmwn 兺 ␾*k 共xn兲␾⬙k共xm兲.
2␮
k=1
um共x兲 = 兺 c̃n␾n共x兲.
共10兲
n=1
Because the basis ␾n is orthonormal, we have using Eq. 共7兲
共16兲
In most cases, the finite sum in Eq. 共16兲 can be evaluated
once in closed analytical form for the basis functions ␾n.
Given the discrete variable representation of the operators
V̂ and T̂, the energies and wave functions are obtained by
diagonalization of the Hamiltonian matrix, Eq. 共3兲. Let Ei be
the eigenvalue associated with the eigensolution ␺i共x兲
N
␺i共x兲 = 兺 cnun共x兲,
共17兲
n=1
where the cn form the ith eigenvector of the Hamiltonian
matrix. With the aid of Eq. 共7兲, the numerical values of ␺i共x兲
at the grid points xk are readily obtained as
N
N
␺i共xk兲 = 兺 cnun共xk兲 = 兺 cn
n=1
N
共15兲
k=1
n=1
␦nk
=
ck
冑w k 冑w k .
共18兲
In other words, except for weight factors, the eigenvectors of
the Hamiltonian matrix directly give the values of the wave
function at the grid points.
N
c̃n = 具兩␾n兩um典 = 兺 wk␾*n共xk兲um共xk兲
III. BLOCH-DISCRETE VARIABLE
REPRESENTATION FOR PERIODIC POTENTIALS
k=1
N
= 兺 wk␾*n共xk兲
k=1
␦mk
冑w k =
冑wm␾*n共xm兲.
共11兲
We substitute Eq. 共11兲 into Eq. 共10兲 and obtain the desired
basis functions
N
um共x兲 = 冑wm 兺 ␾*n共xm兲␾n共x兲.
共12兲
n=1
Note that the initial basis set, Eq. 共4兲, and the quadrature
rule, Eq. 共5兲, must be consistent.11 For instance, letting x
= xk into Eq. 共12兲 and using Eq. 共7兲 provides a necessary
relation between the weight wk and the values of all basis
functions at the grid point xk
冉兺
N
wk =
n=1
963
兩␾n共xk兲兩
2
冊
−1
.
Am. J. Phys., Vol. 73, No. 10, October 2005
共13兲
Let the potential be a P-periodic function: V共x + P兲 = V共x兲.
Floquet’s theorem15 states that we can find a solution to Eq.
共1兲 of the form
␺␬共x兲 = ei␬xF␬共x兲,
共19兲
where F␬共x兲 has the same periodicity as V共x兲 and ␬ is the
propagation constant. Solutions of the form of Eq. 共19兲 are
called Bloch wave functions.16 Bloch functions are similar to
plane waves exp共ikx兲; the difference is the presence of the
periodic modulation F␬共x兲. However, ␬ is not the same as k:
Whereas k is a good quantum number, ␬ is not because
p̂ = −iប ⳵ / ⳵x does not commute with the Hamiltonian. For
this reason, ប␬ is known as a quasi-momentum.17
Because F␬共x兲 is periodic, it can be expanded in a Fourier
series with the fundamental angular frequency ␻ = 2␲ / P. It
follows that ␺␬共x兲 can be expanded as
Hervé Le Rouzo
963
⬁
␺␬共x兲 = e
i␬x
兺
⬁
c ne
in␻x
n=−⬁
=
兺
cnei共␬+n␻兲x ,
共20兲
n=−⬁
where the cn are coefficients to be determined. In view of the
expansion of Eq. 共20兲, it is natural to build a discrete variable
representation with Bloch basis functions of the form
␾n共x兲 =
ei共␬+n␻兲x
冑P
,
共n = − M,…,− 1,0,1,…,M兲.
共21兲
In Eq. 共21兲, ein␻x is the usual Fourier basis, and ei␬x accounts
for the Bloch factor. It is easy to verify that Eq. 共21兲 defines
a set of N = 2M + 1 orthonormal basis functions over the
range 共0 , P兲. The quadrature adopted for the Bloch discrete
variable representation is the simple rectangular rule
wk = ⌬,
xk = k⌬,
共k = 1,2,…,2M + 1兲
共22兲
with the step size
⌬=
P
.
2M + 1
共23兲
Fig. 1. The Krönig-Penney 共full line兲 and Mathieu 共dashed line兲 potentials
given in Eqs. 共29兲 and 共30兲.
M
共24兲
The sum over k can be evaluated analytically.18 We obtain
冋
Tmm =
M共M + 1兲
ប2 2
␬ + ␻2
3
2␮
Tmn =
ប2
共− 1兲m−nei␬共m−n兲⌬
2␮
冋
⫻ −
册
共25a兲
DVR
Hmn
共␬兲 =
ប2
共− 1兲m−nei␬共m−n兲⌬
2␮
册
cos关共m − n兲⌬⬘兴
i␬␻
+ ␻2
,
sin共共m − n兲⌬⬘兲
2 sin2关共m − n兲⌬⬘兴
共26兲
For the special case of ␬ = 0 and P = 2␲, Eq. 共25a兲 and
共25b兲 give the discrete variable representation obtained in
Ref. 9 for the simple Fourier basis set einx / 冑2␲ over the
range 共0 , 2␲兲, namely,
ប2 M共M + 1兲
Tmm =
3
2␮
cos关␲共m − n兲/共2M + 1兲兴
ប2
.
共− 1兲m−n
Tmn =
2 sin2关␲共m − n兲/共2M + 1兲兴
2␮
共27a兲
共27b兲
Finally, we see that for a periodic potential with angular
frequency ␻ = 2␲ / P, sampled at N = 2M + 1 regularly spaced
points over one period, the Bloch discrete variable representation of the Hamiltonian is
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Am. J. Phys., Vol. 73, No. 10, October 2005
+ ␻2
i␬␻
sin关共m − n兲⌬⬘兴
册
cos关共m − n兲⌬⬘兴
,
2 sin2关共m − n兲⌬⬘兴
共28b兲
where m , n = 1 , 2 , … , N, and ⌬ = P / N and ⌬⬘ = ␲ / N.
The N ⫻ N Hamiltonian matrix HDVR共␬兲 in Eq. 共28a兲 and
共28b兲 can be readily computed in terms of ␬. We emphasize
that the nature of the basis sets, the original Bloch functions
and their localized analogs, is transparent in Eq. 共28a兲 and
共28b兲, and is no longer required because the eigenvectors of
HDVR共␬兲 are the amplitudes of the wave functions at the grid
points up to the factor 1 / 冑⌬ 关Eq. 共18兲兴.
A. The Krönig-Penney and Mathieu potentials
where
␲
.
2M + 1
冋
⫻ −
共28a兲
IV. APPLICATION TO MODEL POTENTIALS
共25b兲
⌬⬘ =
册
M共M + 1兲
ប2 2
␬ + ␻2
+ V共m⌬兲
3
2␮
Because the modulus of any Bloch basis function is equal to
1 / 冑P, Eq. 共13兲 gives the weights wk = ⌬ in accordance with
Eq. 共22兲.
We substitute the Bloch basis functions of Eq. 共21兲 and the
quadrature parameters of Eq. 共22兲 into Eq. 共16兲 and obtain
1
ប2
Tmn =
兺 共␬ + k␻兲2ei共␬+k␻兲共m−n兲⌬ .
2␮ 2M + 1 k=−M
冋
DVR
Hmm
共␬兲 =
There are very few periodic potentials for which analytical
solutions are available. The simplest is an infinite array of
rectangular wells, the Krönig-Penney potential.19 The sinusoidal Mathieu potential is more physical 共the potential and
its derivatives are continuous兲, but the calculation is more
involved.20
In this section, we treat both the Krönig-Penney and
Mathieu potentials on the same footing, and show that they
contain the same basic physics. Both potentials have period
P and oscillate between 0 and 2Vm if we take
VKP共x兲 =
再
2Vm − P/4 ⬍ x ⬍ P/4
0
P/4 ⬍ x ⬍ 3P/4
冎
共29兲
for the Krönig-Penney potential, and
冉 冊
V M 共x兲 = Vm + Vmcos
2␲x
P
共30兲
for the Mathieu potential. These potentials are depicted in
Fig. 1.
The dispersion relation, E = E共␬兲, is one of the most important properties of a material; it is the key link between the
crystal structure and its physical properties. Even in one diHervé Le Rouzo
964
Fig. 2. Exact dispersion curves E = Ei共␬兲 and discrete variable representation
values 共circles兲 for five equally spaced ␬ values in the first Brillouin zone.
The horizontal dashed lines are the upper bounds 共2Vm兲 of the potentials.
The parameters are P = 5 bohr and Vm = 1.5 hartree.
mension, E共␬兲 exhibits many features of energy bands in
actual periodic lattices. In the reduced representation that is
adopted here 共␬ 苸 关0 , ␲ / P兴兲, E共␬兲 is a multi-valued function
of ␬.
For the Krönig-Penney and Mathieu potentials, the relation between E and ␬ is a transcendental equation of the
form cos ␬ P = f共E , P , Vm兲, where f is a complicated function
共an infinite determinant for the Mathieu equation兲. However,
it is possible to solve these relations numerically to obtain
the exact energies Ei, one in each band, for a given propagation constant ␬ and a band index i. The associated wave
functions will be denoted as ␺␬i共x兲.
The same potential parameters will be used throughout,
namely, P = 5 bohr and Vm = 1.5 hartree: All values are given
in atomic units.21 Correspondingly, the reduced zone for ␬ is
关0 , ␲ / P兴 = 关0 , 0.628兴 a.u.
The exact dispersion curves, E = Ei共␬兲, in the reduced zone
scheme are given in Fig. 2 for the Krönig-Penney and
Mathieu potentials, for band indices ranging from i = 1 to i
= 6. These dispersion curves19,20 will be used to check the
discrete variable representation results. The dispersion
curves exhibit forbidden energy bands, separating regions of
allowed energy where the solutions ␺␬i共x兲 remain bound as
x → ± ⬁. Both potentials support two narrow, low energy,
bands. For further comparison with the discrete variable representation results, the exact energies for ␬ = 0.2 a.u. and the
same potential parameters are given in Table I.
Table I. Comparison of exact and discrete variable representation values of
the energies 共hartree兲 of the Bloch states with ␬ = 0.2 a.u. in the four lowest
bands for the Mathieu and Krönig-Penney potentials. The discrete variable
representation calculations are done with N = 51 and N = 201 points, respectively. The potential parameters are P = 5 bohr and Vm = 1.5 hartree.
Mathieu potential
Krönig-Penney potential
Energy Ei
Exact
DVR error
Exact
DVR error
E1
E2
E3
E4
0.714 78
2.039 68
2.896 48
4.294 67
2.456⫻ 10−7
5.291⫻ 10−6
1.782⫻ 10−5
5.339⫻ 10−5
0.442 13
1.693 99
3.106 00
4.324 94
0.003 22
0.011 14
0.009 44
0.006 86
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Am. J. Phys., Vol. 73, No. 10, October 2005
Fig. 3. Exact density probabilities 兩␺␬i共x兲兩2 over two periods of the Mathieu
and Krönig-Penney potentials for ␬ = 0.2. The baselines are drawn at the
exact energies Ei given in Table I. The potentials are plotted to help visualize the vertical energy scale.
In Fig. 3 we present the exact density probabilities
兩␺␬i共x兲兩2 of the Mathieu and Krönig-Penney potentials for ␬
= 0.2 a.u. As expected, the states ␺␬1 and ␺␬2, belonging to
the two lowest narrow bands, correspond to levels that are
close to that of an isolated well. Higher states still feel the
presence of the potential wells, but for increasing energy,
they tend toward the traveling wave ei␬x, for which the density probability is constant everywhere.
For discrete variable representation calculations a real
value of ␬ is typically chosen initially, the same for all basis
orbitals. Then the set of associated energies Ei is obtained
from a single diagonalization. More precisely, the steps are
as follows:
1. Choose an odd number N = 2M + 1 of basis functions. This
choice determines N grid points xk according to Eq. 共22兲.
2. For a given ␬ in the reduced zone, construct the N ⫻ N
Hamiltonian matrix HDVR共␬兲 according to Eq. 共28a兲 and
共28b兲. To do so requires the computation of V共xk兲, where
V is given by Eq. 共29兲 or Eq. 共30兲.
3. Diagonalize the discrete variable representation Hamiltonian matrix to obtain the energies Ei and the corresponding values of the wave functions ␺i共xk兲 on the grid.
We chose N = 51 grid points for the discrete variable representation calculations for the Mathieu potential, and N
= 201 for the similar calculations for the Krönig-Penney potential. The errors in Ei are reported in Table I. We see that
the discrete variable representation energies are excellent for
the Mathieu potential, but less precise for the Krönig-Penney
potential in spite of a larger grid. As is well known in the
theory of Fourier series, the reason for this discrepancy is
that discontinuous potentials require more harmonics, which
means more discrete variable representation points are
needed to represent the wave functions near the discontinuities.
For the dispersion curves, the discrete variable representation calculations are done for five equidistant ␬ values spanning the reduced zone as shown in Fig. 2. We see that there
is good agreement between the exact dispersion curves and
the discrete variable representation energies.
We now compare the discrete variable representation wave
Hervé Le Rouzo
965
Fig. 5. The soft Coulomb potential in Eq. 共31兲. The parameters are P
= 10 bohr and Q = a = 1 a.u.
Fig. 4. Wave function moduli 兩␺␬i共x兲兩: Exact 共solid curves兲 and discrete
variable representation 共circles兲 for ␬ = 0.2. The discrete variable representation calculations are done with N = 51 points for the Mathieu potential, and
with N = 201 points for the Krönig-Penney potential 共in the latter case, only
one third of the points are plotted兲.
functions with the exact results.19,20 Except at the band
edges, where the solutions are periodic standing waves 共the
famous Mathieu’s functions when V = V M 兲, the wave functions of the Bloch form, Eq. 共19兲, oscillate aperiodically because the exp共i␬x兲 factor modulates the periodic function
F␬共x兲 as we move from one cell 共or period兲 to the next.
However, the modulus of the Bloch functions 兩␺␬共x兲兩
= 兩F␬共x兲兩 is the same in every cell: All that is modulated is the
phase of the wave functions. In Fig. 4 the exact and discrete
variable representation values of 兩␺␬i共x兲兩 for the three lowest
states are plotted for ␬ = 0.2 a.u. The agreement between the
exact 共curves兲 and discrete variable representation 共circles兲
wave functions is very good for the Mathieu potential. For
the Krönig-Penney potential, a larger grid size is needed to
obtain a similar agreement.
B. The soft Coulomb periodic potential
V共x兲 = −1 / 共1 + x2兲1/2 supports bound and free states similar to
those of the true Coulomb potential.22 We shall take Q = a
= 1 and P = 10 共in atomic units兲. The corresponding periodic potential is depicted in Fig. 5. The atomic energies of
the bound states of the isolated atom22 are reported in
Table II, along with the band edges computed by the
Bloch discrete variable representation with N = 101 grid
points. The discrete variable representation band structure
is given in Fig. 6.
In Fig. 7 the discrete variable representation density probabilities 兩␺␬i共x兲兩2 for ␬ = ␬max / 2 ⯝ 0.157 a.u. are presented.
The states ␺␬1 and ␺␬2, belonging to the two lowest narrow
bands, correspond to levels that are close to those of an isolated well. The third state has a significant density probability outside the wells, that is, between atoms: This state belongs to the conduction band.
V. CONCLUSION
We have shown that the discrete variable representation
with underlying Bloch basis functions is suitable for the calculation of band structures. The method can be applied to an
arbitrary one-dimensional periodic potential. The parameters
are the 共odd兲 number N of grid points and the propagation
constant ␬. The Hamiltonian matrix is computed using Eq.
To illustrate the capabilities of the Bloch discrete variable
representation, we consider a potential whose band structure
cannot be solved analytically. We construct a P-periodic soft
Coulomb potential with
V共x兲 = −
Q2
共31兲
冑a2 + 共x − P/2兲2 ,
for 0 艋 x 艋 P. Such quasi-Coulombic potentials have been
used in multiphoton physics because the atomic potential
Table II. Energies for the soft Coulomb potential in Eq. 共31兲 with parameters Q = a = 1 a.u. For the isolated atom, the exact values are from Ref. 22;
for the crystal with period P = 10 bohr, the edges of the four lowest bands
were computed with N = 101 points.
Atomic energies
Crystal band edges
Energy Ei
Exact
Lower
Upper
E1
E2
E3
E4
−0.669 82
−0.274 94
−0.151 49
−0.092 70
−0.670 28
−0.296 20
−0.173 92
0.023 033
−0.669 29
−0.246 15
0.004 753 4
0.345 14
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Am. J. Phys., Vol. 73, No. 10, October 2005
Fig. 6. The dispersion curves E = Ei共␬兲 for the soft Coulomb potential calculated by the Bloch discrete variable representation with N = 101 points.
Hervé Le Rouzo
966
rials with general anisotropy,” Phys. Rev. B 69, 094301-1–10 共2004兲.
See any elementary quantum physics text such as K. Gottfried, Quantum
Mechanics 共W. A. Benjamin, New York, 1966兲.
5
H. Le Rouzo, “Variational R-matrix method for quantum tunneling problems,” Am. J. Phys. 71, 273–278 共2003兲, and references therein.
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In the context of Lagrange interpolation, Eq. 共7兲 is sometimes called the
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共1986兲.
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R. M. Whitnell and J. C. Light, “Symmetry-adapted discrete variable
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16
Note that because Eq. 共1兲 is linear and second order, two independent
solutions can be found for a given energy E共␬兲 : ␺␬共x兲 and ␺␬共−x兲 are the
independent solutions, unless ␬ P = N␲, where N is an integer. In the latter
case, the two solutions become identical and represent standing waves
共the Mathieu functions for V = V M 兲 corresponding to the band edges.
17
Note that we can derive the exact relation for the electron velocity in
Bloch states: v = 具p̂典 / ␮ = ⳵E共␬兲 / ⳵ប␬. This result should be compared with
the velocity v = 具p̂典 / ␮ = បk / ␮ for free electrons.
18
The calculation is tedious and follows the derivation suggested in Ref. 9.
19
R. A. Smith, Wave Mechanics of Crystalline Solids 共Chapman and Hall,
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20
N. W. McLachlan, Theory and Application of Mathieu Functions 共Clarendon Press, Oxford, 1951兲; M. Abramowitz and I. A. Stegun, Handbook
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21
In atomic units: ␮ = ប = 兩e兩 = 1. Atomic units of length and energy are
1 bohr= 0.529 177 Å and 1 hartree = 27.211 65 eV, respectively.
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J. H. Eberly, Q. Su, and J. Javanainen, “High-order harmonic production
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R. G. Littlejohn and M. Cargo, “Multidimensional discrete variable representation bases: Sinc functions and group theory,” J. Chem. Phys. 116,
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4
Fig. 7. Discrete variable representation density probabilities 兩␺␬i共x兲兩2 over
two periods of the soft Coulomb potential for ␬ = ␬max / 2 with N = 101 points.
The energies of the states 共in hartree兲 are E1 = −0.669 79, E2 = −0.278 14,
E3 = −0.107 73, and E4 = 0.166 27.
共28a兲 and 共28b兲 and then diagonalized. The eigenvalues En
are the desired energies, one in each band n. Another attractive feature is that the components of the nth eigenvector
directly give the values of the nth wave function at the grid
points.
The generalization of the Bloch discrete variable representation to two and three dimensions is simple if the Cartesian
products of one-dimensional basis functions are taken to be
the basis.9 The procedure for creating more general multidimensional discrete variable representations has been recently
addressed.23
ACKNOWLEDGMENTS
It is a pleasure to thank Dr. Eric Charron, Dr. Andrew
Mayne, and Dr. Georges Raseev of the Laboratoire de Photophysique Moléculaire in Orsay, for stimulating discussions
and helpful comments on the manuscript.
a兲
Electronic address: herve.le-rouzo@ppm.u-psud.fr
C. Kittel, Introduction to Solid State Physics 共Wiley, New York, 1996兲,
7th ed., Chap. 7.
2
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S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng,
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acoustic waves in two-dimensional phononic crystal consisting of mate1
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Am. J. Phys., Vol. 73, No. 10, October 2005
Hervé Le Rouzo
967