Full text of DOI:10.1007/s10053-000-8805-1

Eur. Phys. J. D 8, 165–168 (2000)
THE EUROPEAN
PHYSICAL JOURNAL D
EDP Sciences
Societ a` Italiana di Fisica
Springer-Verlag 2000
ꢀ
c
A one-dimensional lattice model for a quantum mechanical free
particle
a
A.C. de la Torre and A. Daleo
Departamento de F ´ı sica, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina
Received 25 March 1999 and Received in final form 20 August 1999
Abstract. Two types of particles, A and B with their corresponding antiparticles, are defined in a one-
dimensional cyclic lattice with an odd number of sites. In each step of time evolution, each particle acts
as a source for the polarization field of the other type of particle with nonlocal action but with an effect
¯
¯
¯
¯
¯
decreasing with the distance: A → · · · BBBBB · · · ; B → · · · AAAAA · · · . It is shown that the combined
distribution of these particles obeys the time evolution of a free particle as given by quantum mechanics.
PACS. 03.65.Bz Foundations, theory of measurement, miscellaneous theories (including Aharonov-Bohm
effect, Bell inequalities, Berry’s phase) – 03.65.Ca Formalism
¯
The modeling of physical reality by means of fictitious amount τ, the particles of type A create antiparticles B
particles that move and react in a substrate of differ- in the same site, particles B in the first neighboring sites,
¯
ent geometrical structures has been a fruitful strategy B in the second neighboring sites and so on. In a similar
¯
that has extended our analysis capabilities beyond the do- way, particles B create particles A and A
main associated with differential equations [1]. The par-
¯
¯
¯
ticles involved in these models are classical in the sense
that they are given precise location and velocity. This
is clearly inadequate for the modeling of quantum sys-
A −→ · · · B B B B B · · ·
¯
¯
B −→ · · · A A A A A · · ·
(1)
tems that require, not only the indeterminacies imposed The same reactions occur exchanging particles and an-
by Heisenberg’s principle, but also nonlocal correlations tiparticles. This creation process extends to the right and
between commuting observables, suggested by the Einstein left of each site up to the two opposing sites in the cir-
Podosky Rosen argument [2] and empirically established cle. Since N is odd, in these two sites particles of the
in the violation of Bell inequalities [3,4]. However this does same sign (either particles or antiparticles) are created.
not forbid the modeling of quantum systems if we do not The number of particles or antiparticles created decreases
2
identify the particles of the model with the quantum parti- with the distance d roughly like 1/d for a distribution of
cles. It is possible, as will be seen in this work, to associate particles confined in a small region within a large lattice
the real quantum particle to a combined distribution of as will be precisely stated later. Before we write the mas-
two types of fictitious particles with nonlocal interaction. ter equation for the time evolution, we can notice some
This simple example model can be trivially extended to qualitative features of the process. It is easy to see that
higher dimensions of space and to a higher number of non- the process has diffusion. If we start, for instance, with
interacting quantum particles and it provides a new point some number of A particles in one site, after two time
¯
of view to study the peculiarities of quantum mechanics. steps, some A antiparticles have been created at the same
Let us assume a one-dimensional lattice with N sites site reducing the number of A particles, but also some A
on a circle and lattice constant a. We assume N to be particles appear in the first neighboring sites. The net ef-
an odd integer. The reason for this restriction will become fect is diffusion. It is less obvious that, even though the
clear later. The inclusion of even values for N would intro- process has left-right symmetry, we may also have drift to
duce unwanted complications in the model. Each site can the right or to the left. In order to see how this is possi-
be occupied by any number of particles of type A, B or ble we notice that A particles reject B particles from the
¯
¯
¯
by their corresponding antiparticles A, B. Particles and site because B are created there, whereas B particles at-
antiparticles of the same type annihilate in each site of tract neighboring A particles to his site. Therefore if we
the lattice leaving only the remaining excess of particles have an asymmetric configuration like AB, the center of
or antiparticles of both types A and B. At each time step, the combined distribution will move towards B. The drift
t → t + 1, corresponding to a time evolution by a small direction and velocity is then encoded in the shape and
relative distribution of both types of particles. We will
see that, although the distribution of particles are widely
a
e-mail: dltorre@mdp.edu.ar

166
The European Physical Journal D
distorted after few time steps, the drift direction and ve- time evolution of the process is the sum of the square of
locity remain invariant. the number of A particles (or antiparticles) plus the sum
A convenient way to label the sites of the circular lat- of the square of the number of B particles (or antiparti-
tice is by an index s running from −L to L. Since the num- cles). This approximately conserved quantity can be used
ber of sites N = 2L+1 is odd, the index s will be integer. for normalization and can be given a physical meaning
It is of course irrelevant which site has the label s = 0. like energy density or probability density. It is therefore
2
s
2
s
Let as(t) and bs(t) be the number of particles of type A relevant to define a combined distribution a (t)+b (t), as-
and B respectively at the site s at time t, normalized in a sociated to this approximately conserved quantity. We will
way that will be specified later (anyway the master equa- see that the drift velocity of this combined distribution,
tion is independent of the normalization). When as(t) or given by
bs(t) take negative values they denote the number of an-
X
tiparticles. At a particular site of the lattice, the number
of particles change as particles or antiparticles are created
in it by the particles in other sites. The time evolution of
the process is then defined by the equations
hV i = 4g
a (t)b (t)G(s − r),
(5)
s
r
s,r
is also approximately conserved in the time evolution.
Given a distribution {as(t), bs(t)}, we can change the drift
velocity by an amount v, without changing the shape of
the combined distribution by means of the local transfor-
mation
L
X
2
as(t + 1) = as(t) + τg
b[s+d](t)F(d)
a[s+d](t)F(d),
d=−L
L
X
2
0
bs(t + 1) = bs(t) − τg
(2)
as(t) = as(t) cos(vas/2) − bs(t) sin(vas/2)
0
d=−L
b (t) = a (t) sin(vas/2) + b (t) cos(vas/2).
(6)
s
s
s
where the square brackets in the index, [s + d], denotes
modulo N”, that is, with a value in the closed inter-
val [−L, L]; g is related to the lattice constant a by g =
2π)/(Na) (it corresponds to the reciprocal lattice con-
Both quantities mentioned are “approximately” and not
strictly conserved due to the discrete nature of the model.
We will see that taking smaller time steps τ leads to
better conservation. These features have been checked in
a computer simulation of the process. A circular lattice
with N = 801 sites (L = 400) and with lattice con-
stant a = 1 was chosen. Several shapes of initial distri-
butions were tried: Gaussian, uniform and random, with
“
(
2
2
stant); τ is a time scale small enough to make τg N ꢀ 1,
and the function of the distance F(d) is defined as
L
X
1
N

2π
2
i
kd
N
F(d) =
k e
several widths and drift velocities. The time dependence
P
k=−L
2
2
s s
of M(t) =
(a (t)+b (t)) and of the drift velocity given
s
cos(πd/N)
−3


d
in equation (5) was studied. Taking a time step τ = 10
we found that these quantities remain constant after t =
,
(−1)
if d = ꢁ1, ꢁ2, ..., ꢁL .
2
2
2
sin (πd/N)
=
−5


1
2
1000 time steps, with a relative variation less than 10
4
(N − 1)
if d = 0
−
1
for the Gaussian case, 4 × 10 for the uniform distribu-
tion, and 0.04 for the random distribution. For larger time
steps, τ = 0.005, these quantities remain constant (less
than 1% relative variation) for the Gaussian and uniform
case at t = 1000 but the random case begins to show sig-
nificant departure from constancy. At τ = 0.010 only in
the Gaussian case these quantities remain constant (less
than 0.1%). The time evolution of the shape of the com-
bined distribution is strongly reminiscent of the time evo-
(3)
For later use we define a similar function G(d) as:
L
X
i
N
2π
N
i
kd
G(d) =
ke
k=−L

d
(
−1)

=
2 sin(πd/N)
if d = ꢁ1, ꢁ2, ..., ꢁ2L .
if d = 0
(4) lution of quantum mechanical wave packets. For instance,
a Gaussian distribution for A and B particles, modified by
equation (6) in order to have drift, will evolve increas-

0
The alternating sign in the definition of F(d) corresponds ing the width and drifting but maintaining the Gaussian
to the fact that particles and antiparticles are created at shape. A uniform distribution will develop side lobes in
alternating sites, and the different sign in equation (2) the evolution. A remarkable feature is that the process
is due to the difference in the role of particle and an- smooths out the random fluctuations of an initial distri-
tiparticle in relation (1). If the particles are confined in bution.
a small region within a large lattice, the main contribu-
The resemblance of the process with quantum mechan-
tion in the sums of equation (2) comes from terms with ics is striking. We will indeed show that the process here
2
distance |d| ꢀ N. In this limit we have |F(d)| ≈ 1/d as defined corresponds to a quantum mechanical free particle
mentioned before.
in a lattice. An extensive numerical simulation of the pro-
The number of A or B particles are not conserved in cess is therefore not necessary because quantum mechan-
the time evolution. Neither is the sum of particles con- ics provides a faithful representation for it. Let us define
served. A quantity that is approximately conserved in the then an N-dimensional Hilbert space spanned by a basis

A.C. de la Torre and A. Daleo: A one-dimensional lattice model for a quantum mechanical free particle
167
{
ϕs} s = −L, −L + 1, · · · , L corresponding to the eigen- We have then
vectors of the position operator X. Then, Xϕs = asϕs.
In this finite dimensional Hilbert space, we can not define
the momentum operator P by means of the usual com-
mutation relation. The alternative way to define P is to
choose first an unbiased basis [5–7] {φk},
L
X
2
cs(t + 1) = cs(t) − iτg
cr(t)F(s − r).
(14)
r=−L
Reordering the terms in the sum and using the “modulo
N” notation, we get
L
X
1
2π
N
i
ks
φk = √
e
ϕs,
(7)
L
X
N
2
s=−L
cs(t + 1) = cs(t) − iτg
c[s+d](t)F(d).
(15)
d=−L
and with it, we define the momentum by the spectral de-
composition
Finally if we explicitly write the coefficients with real and
L
imaginary part, cs(t) = as(t) + ibs(t), we get equation (2)
X
above.
P =
gkφkhφk, ·i
P
2
We can here check that M(t) =
imately conserved
|cs(t)| is approx-
k=−L
s
X
1
2π
=
gke k(s−r)ϕshϕr, ·i.
i
N
(8)
N
k,s,r
M(t + 1) = M(t)
X
X
2
4
∗
The momentum eigenvalues and the relative phases to
build the basis {φk} have been chosen such that P is the
generator of translations. That is, with this choice, the
operator Ua = exp(−iaP) is such that Uaϕs = ϕs+1.
The translation is cyclic at the border, UaϕL = ϕ−L. If
we had taken N even, the right hand side of this equa-
tion should have a minus sign. This would complicate
the model of relation (1) introducing a change of sign at
some appropriated places. In order to have a simple lattice
model for the quantum free particle we prefer to restrict
ourselves to odd values of N.
+ τ g
cr(t)c (t)
F(r − s)F(s − u). (16)
u
r,u
s
The term linear in τ vanishes because the symmetric func-
tion F appears multiplied by an anti-symmetric factor. We
see here that the “derivative” (M(t + 1) − M(t))/τ van-
ishes like τ in agreement with the numerical simulation of
the process. We can write the functions F in their sum-
mation representations and, performing the sum over s,
we get
L
X
X
F(r − s)F(s − u) = 1
k e
2π
N
(17)
The state of a free quantum particle, given by
4
i
k(r−u)
.
N
L
s
k=−L
X
Ψ(t) =
cs(t)ϕs,
(9)
This sum can be evaluated as was done in equations (3,
) but we don’t need it. In the limit N ꢂ d = r − u, that
s=−L
4
will change according to the time evolution operator (we is, when the particles are confined in a small region of a
set ~ = 2m = 1)
large lattice, we get
ꢀ
ꢁ
2
π4
X
Ut = exp(−iP t).
(10)
(r−u)
(r−u)2
2
∗
(−1)
M(t+1)=M(t)+τ
2
cr(t)c (t)
+M(t) ·
u
5
Let us consider the evolution of the coefficients of the ex-
pansion given in equation (9), in one step of discretized
time: t0 = τt and t1 = τ(t + 1) with a small time scale τ
and t positive integer. We have
r
6=u
(18)
A similar result is obtained for the drift velocity, propor-
tional to the expectation value of P
L
X
X
∗
cs(t + 1) =
cr(t)hϕs, Uτ ϕri.
(11)
hPit = −ig
c (t)cr(t)G(s − r).
(19)
s
r=−L
s,r
2
For τ small enough such that τkP k ꢀ 1, that is τ ꢀ
In terms of as(t) and bs(t) this equation becomes the equa-
2
(
a/π) , the time evolution operator can be linearized and
tion (5) above. Here again, considering the time evolution
we obtain
hPit+1, the term linear in τ vanishes because it contains
P
L
[F(u − s)G(s − r) − G(u − s)F(s − r)] which is zero
X
s
2
as can be calculated with the summation representation
cs(t + 1) = cs(t) − iτ
cr(t)hϕs, P ϕri.
(12)
(13)
of the functions F and G. We obtain then
r=−L
2
5
Using equation (8) we calculate the matrix element
hPit+1 = hPit − iτ g
X
X
∗
L
×
c (t)c (t)
F(u − s)G(s − r)F(r − v).
X
u
v
1
2π
N
2
2
2
i
k(s−r)
.
hϕs, P ϕri = g
k e
u,v
s,r
N
(20)
k=−L

168
The European Physical Journal D
showing that the drift velocity is constant to order τ, that general structure of the process shown in relation (1) re-
is, the “derivative” vanishes with τ in agreement with the mains unchanged but the function F of equation (2) will
numerical simulation of the process. Finally, applying a not be given by equation (3) but will have to be calculated
boost transformation exp(iXv/2) to the state of equa- from an appropriate time evolution operator. The process
tion (9), we prove equation (6).
can also be extended to two or three space dimensions
The one-dimensional lattice model here presented pro- but with larger computer requirements for the numerical
vides a simple representation for the position and momen- simulations.
tum of a free quantum mechanical particle. In this model
Since the advent of quantum mechanics, there have
we require that N should be odd. Let us see what hap- been numerous attempts to develop a classical image for
pens in the case where N is even. In this case, the model quantum behavior. For the reasons already mentioned
evolves according to equation (2) with the summations at the beginning, the attempts in terms of particles are
running from −N/2 to N/2 and with the same function doomed. The model here presented, besides being a “di-
F(d) defined in equation (3). Notice that this function vertissement” in theoretical physics, also suggests the pos-
vanishes at the extreme values of d, that is F(ꢁN/2) = sibility of a classical image for quantum mechanics in
0
. This model can be interesting in itself but it is no terms of two fields A(x, t) and B(x, t) where each field
longer equivalent to the quantum mechanical system. acts as a source for the polarization of the other. Indeed,
The connection is lost in the step from equation (14) to in the continuous limit, the particles and antiparticles of
equation (15). For the cases when the argument s − this work become creation and destruction quanta of two
r of the function F in equation (14) take values ex- fields that turn out to be the real and imaginary part of
ceeding N/2, we should introduce a minus sign if we the wave function; Ψ(x, t) = A(x, t)+iB(x, t). The contin-
want to change the argument to d as in equation (15) uous extension of equations (2, 3) provide the equations
(
in the case N odd, no sign change is needed). The of motion for the fields that are equivalent to Schr o¨ dinger
reason for this change can be traced to the change equation. This continuous study has been done somewhere
in sign produced by the translation operator when else [8] and suggests an interpretation of quantum mechan-
the site labeled by ꢁL is crossed as mentioned after ics as a classical field theory, not more weird than classical
equation (8). It would be possible to include even val- electrodynamics. These considerations may provide a new
ues for N but at the cost of complicating the model. For point of view for studying the peculiarities of quantum
this we would have to change the rules of relation (1) ex- mechanics.
changing particles and antiparticles when we cross the site
with label ꢁL. These complications are unwanted and we
We would like to thank H. M ´a rtin for discussions and com-
ments. This work has been done with partial support from
prefer to accept the fact that position and momentum of a
quantum mechanical particle can be easily modeled only “Consejo Nacional de Investigaciones Cient ´ı ficas y T ´e cnicas”
with a cyclic lattice with an odd number of sites. In the (CONICET), Argentina (PIP grant No. 4342/96). One of us
case of a quantum particle confined in a very small region (A.D.) would like to thank the “Comisi o´ n de Investigaciones
(
say, 10 sites) of a very large lattice (say, close to one mil- Cient ´ı ficas” (CIC) for financial support.
lion sites) it doesn’t matter whether N is even or odd for
all times until, due to drift or diffusion, the distribution
reaches the sites with label close to ꢁN/2. However for References
small lattices and for extended distributions it does mat-
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ter, and only in the odd N case the model of relation (1)
describes a quantum mechanical particle. This is a further
indication of the essential nonlocal character of quantum
mechanics. There is another case in quantum mechanics
where an even or odd number of states has important
qualitative consequences. This is in finite dimensional real-
izations of angular momentum. Whereas intrinsic angular
momentum, the spin, of a particle can have an even or odd
number of states, the orbital angular momentum, arising
from position and momentum, can only have a realization
with an odd number of states.
Barab ´a si, H.E. Stanley, Fractal Concepts in Surface
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. W.K. Wooters, Ann. Phys. 176, 1 (1987).
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The model presented can be extended from the free
particle to the case of a position dependent potential. The