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Some observations on quantum mechanics in disordered systems
Chaos, Printed
Solironr & Fractaols Vol. 3, No. 2, pp. 203-209, in Great Britain
Some Observations
1993 0
on Quantum Mechanics in Disordered Systems MASSIMILIANO
Dipartimento
di Ingegneria
0960.0779/93$6.00 + .oO 1993 Pergamon Press Lfd
Chimica,
Facolta
GIONA
di Ingegneria, Universita Roma, Italy
(Received
25 August
“La Sapienza”,
Via Eudossiana
18, 00184
1992)
Abstract-The localization principle for vibrations on a percolation cluster and, more generally, on disordered systems leads to a modification of the Schrodinger equation by the addition of one term depending on the characteristic relaxation related to anomalous diffusion. As a consequence, quantum particles in disordered structures are subjected to an extra potential level resulting from the heterogeneous topology and geometry. This phenomenon is illustrated by a boundary condition problem at the interface between Euclidean and disordered media. A time dependent Schrodinger equation involving fractional operators is derived from the operational calculus of quantum mechanics, and from the dependence of the relaxation length on frequency.
Apart from some important but specific topics, research in the field of disordered systems has produced three major results generally applicable in all the physical sciences: the awareness that disordered structures cannot be treated by means of a homogeneous formalism, implying that geometry and topology modify dynamical features [l-3]; the discovery of anomalous diffusion [4-51; and the analogy between vibrations and diffusion, leading to the concept of fractons [6-81. In this paper the term disordered system is used to indicate all the physical structures for which the anomalous diffusion exponent D, is greater than 2. The first relevant consequence of this analogy is the localization principle [9], which can be stated as follows: in the absence of potentials, the solution of the Schrodinger equation on a disordered system deviates from the planar wave solution because of the presence of a decaying exponential term ~(r,
w) = A exp [-a(c(Dw)~/4D~l,
(1)
where D, because of u = D,/( D, - 1). In equation frequency w and
=
MD, u + D,
(l),
c(D,+,) = cos(~~/D,)
-1
A is the localization f isin(n/D,)
= Zk,
length
depending
on the
(3)
where the sign indicates the propagation direction. The paper develops some observations regarding the consequences of the localization principle. The main goal is to express the implications of equation (l)-(3) analytically. A modified Schrodinger equation for disordered media (in the sense discussed above) is therefore derived and some of its physical consequences discussed. In particular, the analysis is presented of a boundary condition problem in a junction formed by a Euclidean and by a disordered system.
M. GIONA
204
It is worth noting that all the macroscopic equations derived in this article should be regarded as approximations. Given the present level of knowledge on disordered and fractal media, it is impossible to describe the structure of these media by means of a local set of coordinates, and to express local balance and field equations through this set of coordinates. Nevertheless, this level of approximation is extremely important: all the information on the probability density function for diffusion or on the solution of the Schrodinger equations is in this sense approximate since r is the usual radial Euclidean coordinate [lo, 111. Moreover, the author believes that the analytic-albeit approximate and averagedapproach to the dynamics of disordered systems, which is not limited to scaling analysis but seeks to develop field equations, could clarify many of the physical implications of disorder, thus leading to new theoretical predictions and opening up a channel of communication between researchers engaged in different areas of scientific investigation. Comparison of equation (1) with the Euclidean planar wave solution (DW = 2) r/%(r, m) = where
p = [2mE]“*,
Aexpbrlhl,
allows the identification a -=A
Equation
(4)
(1) can therefore
be written
P
h’
(5)
as
W(r, 0) = Aexp[-pZ:,r/h]. Analysis is here focused on the one-dimensional approximation, i.e. r = 1x1. On the assumption that equation (1) represents the solution of the Schrodinger in a disordered medium, it is natural to assume the wave equation of the form
(6) equation
*+2nz+k*y,=0. dr* The constants (Y and k can be obtained by positing the eigenvalues of the characteristic where v = cos (rr/D_.): equation associated to (6), A = -_(y + i[k2 - a?]1’2, equal to -pXihi
k2=F.
2mE h’
wave equation. For this purpose, the The next step is to derive the time-dependent length 13.to the crucial equation is represented by the scaling law relating the relaxation frequency w and consequently to the energy E. Equation (1) comes from the Laplace inversion of the equation linking the average probability density function for diffusion ( P(r, t)) to the solution of the wave equation w,
PI (P(r,
t))
= I,“n(w*)W(r,
o*)exp(-w*2t)dw*,
(9)
where n(w*) is the density of states. According to equation (9), alit must be adimensional and therefore w** has the dimension of a frequency w, i.e. cc)= w *2. Thus, the relaxation length h = A.(w*) scales with frequency w as [3,8] A-’ _ ,*2/D,,. _ ,J/D,
(10)
Quantum
mechanics
The relaxation length is inversely proportional E may therefore be expressed as
in disordered
205
systems
to the momentum
p = [2mE]“*.
The energy
E ‘I K(D,)(ho)z’Dy
(11)
where the constant K(D,) is a structure parameter of the disordered medium; K(D,) = 1 for D, = 2. Equation (ll), linking energy and frequency in a disordered medium, can be put in operational form. For this purpose it is sufficient to observe that in classical (Euclidean) quantum mechanics E = hw
(12)
and therefore E I&= hwq = H,[qj], HE being the energy operator. be decomposed into Fourier components r+%Ct) = /rP( r, w) exp (-iwt)
Since a wave packet can
do,
the energy operator is nothing but the inverse Fourier transform of Er@, i.e. HE[r&] = F-‘[hw~/j From this observation, and from equation (ll), energy operator can be written as
= iii $.
(13)
it follows that in a disordered
aziD-ly( r, w)
= -(-ih)2’DwK(DW)
3#L
system the
(14)
’
where aq/atq is the fractional operator of order 4 [12]. The last relation can be obtained from the Laplace transform properties of the fractional operators by positing the Laplace variable s equal to -iw. In the general tridimensional isotropic case, equation (6) can be expressed in the form
where r/jr = r&, analogous
and n, is the radial unit vector.
d/Eq
one arrives at the time dependent
-h2
!mV*rjYh
From
+ i(-ih)liDWv[K(D,)]
equation
(14) and from its
$/D, Yj s
wave equation
a’/“wv ’ at r/D,
apI
#Lq,
= -(ih) 2’DwK(D,)
F.
(16)
It should be observed that our derivation of equation (16) employs only scaling arguments, and therefore this equation should be considered correct as long as the scaling law-equation (10) -used in its derivation holds true. It is also worth mentioning that even in the case of the wave equation, the macroscopic field equation describing dynamic phenomena in disordered media can be expressed in which have been recently adopted in describing the terms of fractional operators, complementary phenomenon of anomalous diffusion in disordered systems [13,14]. The
M. GIONA
206
fractional calculus therefore appears to be an appropriate analytical tool for the description of correlations in dynamics on disordered structures. A first interesting application of the above results, and in particular of the more tractable time-independent wave equation (6), is given by the analysis of a boundary condition problem in the presence of a potential step at a junction between a disordered and a Euclidean structure. This situation is shown in Fig. 1. Let the energy of the particles E be less than the potential Vo(E < V,). In the case of Fig. 1, the Schrodinger equations read dr2
+ 2a dr * dr2
+ k;q
= 0, x < 0, (17)
+ k2q = 0, x > 0,
where k$ = 2mEb*, k* = 2m(E - V,)/h* and r = 1x1. Positing B = (Y+ ik+, k+ = [ki - a * ] ‘12, the solution of equation q = Al exp(@)
+ A2exp(B*r),
I/J = Br exp(ikx)
+ B,exp(-ikx),
where the star indicates the complex conjugate. The last equation must satisfy continuity conditions
(17) is given by
x < 0, x > 0,
at x = 0
A, + A2 = B, + B2, /_?A, + B*A2 = ik(B1 leading
- B2),
to B,
=
p-b* -B*
A
2
=
A
p-ik -p*
A +
B*+ik
+
’
+ ik ik
(19)
’
-p*
+
equal
ik
B2’
to zero.
Therefore,
the solution
(_~~+pTk)Alev(-=) =ia+yL: _JAlexP(--Kx). (20) ”
Fig. 1. Schematic
2’
2ik
+
Since V. > E, k = iK and B2 must be identically of the wave equation for x > 0 reads
v=
B
-/?I* + ik
representation
II--
..~_._..____.__
of disordered medium-Euclidean step barrier in the Euclidean
“,
system junction medium.
in the presence
of a potential
Quantum
mechanics
in disordered
Positing qQ= MA 1exp ( --Kx), T = (M12/4, z = V,/E,
sion coefficient,
systems
201
T can be regarded
as the transmis-
one readily obtains
T=(1 -
(1 - v,‘> ql,‘) + (rp -
qz -
(21)
1])2’
which should be compared with the Euclidean solution T = l/z. Given equation (21), if the potential step satisfies the condition V. = E(l + q2) (22) then T = 1. This peculiar effect, which implies a nonmonotonic behaviour of the transmission coefficient T, Fig. 2, is a consequence of disorder, i.e. of Q,f 0. It can be intuitively understood by observing that, due to its anomalous properties in diffusional and vibrational dynamics, a disordered medium acts as a potential barrier for a particle of energy E whose amplitude is E $. In this way, equation (22) can be regarded as the condition under which particles in the disordered medium ‘feel’ the entire x-axis as an equipotential. This phenomenon is intuitive if one observes that an exponent D, greater than 2 implies correlation in the random motion, which in the case of the Schrbdinger equation appears in the form of an equivalent extra potential energy of amplitude Eq2. Since A - p-l - E-@, implying that the relaxation length is shorter for more energetic particles, it is natural to expect the extra potential due to the presence of the relaxation length to be an increasing function of the energy E. The complementary case is shown in Fig. 3 for the transition from a Euclidean to a disordered system in which a potential step is present. As in the previous case, we have
d2w +k&+9=0, x dx2 &JJ dV +2&z+ dr2 which, after some algebraic manipulation, T=
coefficient
(23)
k;t,b=O,x>O,
leads to the following expression for T 1 (24)
1 + (q + V[z + QzJ2 - 11).
1 Fig. 2. Transmission
< 0,
T vs z for
2 the
z
case of Fig. 1: (a) (d) D, = 3.5.
3 D, = 2.0;
(b)
D, = 2.5;
(c)
D, = 3.0;
M. GIONA
208
V
Fig. 3. The complementary
case of that of Fig. 1, when a potential
barrier
is present
in the disordered
system
As in the case of Fig. 1, the disordered medium acts with respect to the particle in the Euclidean medium as an extra barrier of energy Eq?, which is the complementary result of equation (22). To sum up, the presence of anomalous behaviour in diffusion (Dw > 2) leads to highly nontrivial predictions for quantum particles in a disordered medium. These predictions can be unified by means of the hypothesis that temporal correlations in random motion are equivalent to an additional potential whose magnitude is proportional to the energy of the particles. The analysis presented is a first step towards an understanding of anomalous scaling in the dynamics of disordered systems by means of field equations. It seeks to stimulate the formulation of a consistent, even if approximate, theory of anomalous behaviour capable of furnishing predictions or of falsification by experimental results in the other branches of physics. In particular, attention should be drawn to the effect of disordered and fractal interfaces on the structure of surface energy levels, since this could lead to meaningful results in surface physics and in the theory of adsorption. It is important to point out the relationship existing between the present work and the contributions on fractal space-time and fractal ideas in quantum mechanics. It is known [15, 161, that the Hausdorff dimension of a quantum mechanical path (e.g. a quantum oscillator) is equal to 2. From this and from the theoretical arguments (see e.g. Nottale [17]) the idea of a fractal space-time was developed and related to the Heisenberg’s uncertainty principle 118-211. The present analysis does not seek to present new results on the fundamental problem of a Cantor-like model of space-time but rather to derive from scaling arguments developed for disordered fractal systems, some of their implications in terms of classical quantum mechanics. It is possible that some of these results may prove useful in the formulation of a unified theory of quantum mechanics in fractal spaces. However, it should be pointed out that the description of diffusion processes on fractals as well as the result expressed by equation (14) in terms of fractional derivative are the manifestation of the appearance of correlations in the random motion on fractals and are not related to the definition of a fractal time. Finally, it may be pointed out that the modification of the Schrodinger equation, as a result of scaling arguments, and the phenomenon of energy shifting are not surprising since they are ultimately a consequence of the non-Brownian nature (in the sense of usual Wiener processes) of the fluctuations in anomalous disordered media. Acknowledgements-The
author
is grateful
to H. E. Roman
for useful suggestions.
Quantum
mechanics
in disordered
systems
209
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11.
12. 13. 14.
15. 16. 17. 18. 19. 20. 21.
B. B. Mandelbrot, The Fractul Geometry of Nature. Freeman, San Francisco (1982). L. Pietronero and E. Tosatti (eds), Fructals in Physics. North Holland, Amsterdam (1985). A. Bunde and S. Havlin (eds), Fractul and Disordered Systems. Springer, Berlin (1991). P. G. de Gennes, La Recherche 7, 919 (1976). S. Havlin and D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 36, 695-798 (1987). S. Alexander and R. Orbach, J. Phys. Lett. 43, L625 (1982). R. Rammal and G. Toulouse, J. Whys. Len. 44, L13 (1983). H. E. Roman, S. Russ and A. Bunde, Pfrys. Rev. Len. 44, 1643 (1991). R. Orbach, Science 231, 814 (1986). For the numerical analysis of the probability density function for diffusion (P(r, t)) where r is a Cartesian radial coordinate see: H. E. Roman, A. Bunde and S. Havlin, in Reluxation in Disordered Systems and Related Topics, edited by A. Campbell and C. Giovannella. Plenum Press, New York, (1990); H. E. Roman, A. Bunde and S. Havlin, Ber. Bunsenges. Phys. Chem. 93, 1205 (1989). Theoretical contributions on the functional form of (P(r, t)) employing the Cartesian coordinate r have been made by: R. A. Gouyer, Phys. Rev. A 32, 2324 (1985) and by: B. O’Shaugnessy and I. Procaccia, Phys. Rev. Lett. 54, 455 (1985). K. Oldham and J. Spanier, The Fractional CulcuZus. Academic Press, New York (1974). M. Giona and H. E. Roman, A theory of transport phenomena in disordered media, Chem. Eng. J. 49, 1 (1992). M. Giona and H. E. Roman, Fractional diffusion equation on fractals part I, J. Phys. A. 25, 2093-2106 (1992); H. E. Roman and M. Giona, Fractional diffusion equation on fractals Part II, J. Phys. A 25, 2107-2118 (1992). L. F. Abbott and M. B. Wise, Am. J. Phys. 49, 37-39 (1981). E. Campesino-Romeo, J. C. D’Olivo and M. Socolovsky, Phys. Lett. 89A, 321-324 (1982). L. Nottale, Inc. J. Mod. Phys. 4, 5047-51 (1989). G. N. Ord, J. Phys. A 16, 1869-1884 (1983). M. S. El Naschie, Quantum mechanics and the possibility of a Cantorian space-time, Chaos, Solitons & Fructals 1, 485-488 (1991). M. S. El Naschie, On the uncertainty of information in quantum space-time, Chaos, Solitons & Fractuls 2, 91-94 (1992). M. S. El Naschie, Multi-dimensional Cantor sets in classical and quantum mechanics, Chaos, Solitons & Fructals 2, 211-220 (1992).
Solironr & Fractaols Vol. 3, No. 2, pp. 203-209, in Great Britain
Some Observations
1993 0
on Quantum Mechanics in Disordered Systems MASSIMILIANO
Dipartimento
di Ingegneria
0960.0779/93$6.00 + .oO 1993 Pergamon Press Lfd
Chimica,
Facolta
GIONA
di Ingegneria, Universita Roma, Italy
(Received
25 August
“La Sapienza”,
Via Eudossiana
18, 00184
1992)
Abstract-The localization principle for vibrations on a percolation cluster and, more generally, on disordered systems leads to a modification of the Schrodinger equation by the addition of one term depending on the characteristic relaxation related to anomalous diffusion. As a consequence, quantum particles in disordered structures are subjected to an extra potential level resulting from the heterogeneous topology and geometry. This phenomenon is illustrated by a boundary condition problem at the interface between Euclidean and disordered media. A time dependent Schrodinger equation involving fractional operators is derived from the operational calculus of quantum mechanics, and from the dependence of the relaxation length on frequency.
Apart from some important but specific topics, research in the field of disordered systems has produced three major results generally applicable in all the physical sciences: the awareness that disordered structures cannot be treated by means of a homogeneous formalism, implying that geometry and topology modify dynamical features [l-3]; the discovery of anomalous diffusion [4-51; and the analogy between vibrations and diffusion, leading to the concept of fractons [6-81. In this paper the term disordered system is used to indicate all the physical structures for which the anomalous diffusion exponent D, is greater than 2. The first relevant consequence of this analogy is the localization principle [9], which can be stated as follows: in the absence of potentials, the solution of the Schrodinger equation on a disordered system deviates from the planar wave solution because of the presence of a decaying exponential term ~(r,
w) = A exp [-a(c(Dw)~/4D~l,
(1)
where D, because of u = D,/( D, - 1). In equation frequency w and
=
MD, u + D,
(l),
c(D,+,) = cos(~~/D,)
-1
A is the localization f isin(n/D,)
= Zk,
length
depending
on the
(3)
where the sign indicates the propagation direction. The paper develops some observations regarding the consequences of the localization principle. The main goal is to express the implications of equation (l)-(3) analytically. A modified Schrodinger equation for disordered media (in the sense discussed above) is therefore derived and some of its physical consequences discussed. In particular, the analysis is presented of a boundary condition problem in a junction formed by a Euclidean and by a disordered system.
M. GIONA
204
It is worth noting that all the macroscopic equations derived in this article should be regarded as approximations. Given the present level of knowledge on disordered and fractal media, it is impossible to describe the structure of these media by means of a local set of coordinates, and to express local balance and field equations through this set of coordinates. Nevertheless, this level of approximation is extremely important: all the information on the probability density function for diffusion or on the solution of the Schrodinger equations is in this sense approximate since r is the usual radial Euclidean coordinate [lo, 111. Moreover, the author believes that the analytic-albeit approximate and averagedapproach to the dynamics of disordered systems, which is not limited to scaling analysis but seeks to develop field equations, could clarify many of the physical implications of disorder, thus leading to new theoretical predictions and opening up a channel of communication between researchers engaged in different areas of scientific investigation. Comparison of equation (1) with the Euclidean planar wave solution (DW = 2) r/%(r, m) = where
p = [2mE]“*,
Aexpbrlhl,
allows the identification a -=A
Equation
(4)
(1) can therefore
be written
P
h’
(5)
as
W(r, 0) = Aexp[-pZ:,r/h]. Analysis is here focused on the one-dimensional approximation, i.e. r = 1x1. On the assumption that equation (1) represents the solution of the Schrodinger in a disordered medium, it is natural to assume the wave equation of the form
(6) equation
*+2nz+k*y,=0. dr* The constants (Y and k can be obtained by positing the eigenvalues of the characteristic where v = cos (rr/D_.): equation associated to (6), A = -_(y + i[k2 - a?]1’2, equal to -pXihi
k2=F.
2mE h’
wave equation. For this purpose, the The next step is to derive the time-dependent length 13.to the crucial equation is represented by the scaling law relating the relaxation frequency w and consequently to the energy E. Equation (1) comes from the Laplace inversion of the equation linking the average probability density function for diffusion ( P(r, t)) to the solution of the wave equation w,
PI (P(r,
t))
= I,“n(w*)W(r,
o*)exp(-w*2t)dw*,
(9)
where n(w*) is the density of states. According to equation (9), alit must be adimensional and therefore w** has the dimension of a frequency w, i.e. cc)= w *2. Thus, the relaxation length h = A.(w*) scales with frequency w as [3,8] A-’ _ ,*2/D,,. _ ,J/D,
(10)
Quantum
mechanics
The relaxation length is inversely proportional E may therefore be expressed as
in disordered
205
systems
to the momentum
p = [2mE]“*.
The energy
E ‘I K(D,)(ho)z’Dy
(11)
where the constant K(D,) is a structure parameter of the disordered medium; K(D,) = 1 for D, = 2. Equation (ll), linking energy and frequency in a disordered medium, can be put in operational form. For this purpose it is sufficient to observe that in classical (Euclidean) quantum mechanics E = hw
(12)
and therefore E I&= hwq = H,[qj], HE being the energy operator. be decomposed into Fourier components r+%Ct) = /rP( r, w) exp (-iwt)
Since a wave packet can
do,
the energy operator is nothing but the inverse Fourier transform of Er@, i.e. HE[r&] = F-‘[hw~/j From this observation, and from equation (ll), energy operator can be written as
= iii $.
(13)
it follows that in a disordered
aziD-ly( r, w)
= -(-ih)2’DwK(DW)
3#L
system the
(14)
’
where aq/atq is the fractional operator of order 4 [12]. The last relation can be obtained from the Laplace transform properties of the fractional operators by positing the Laplace variable s equal to -iw. In the general tridimensional isotropic case, equation (6) can be expressed in the form
where r/jr = r&, analogous
and n, is the radial unit vector.
d/Eq
one arrives at the time dependent
-h2
!mV*rjYh
From
+ i(-ih)liDWv[K(D,)]
equation
(14) and from its
$/D, Yj s
wave equation
a’/“wv ’ at r/D,
apI
#Lq,
= -(ih) 2’DwK(D,)
F.
(16)
It should be observed that our derivation of equation (16) employs only scaling arguments, and therefore this equation should be considered correct as long as the scaling law-equation (10) -used in its derivation holds true. It is also worth mentioning that even in the case of the wave equation, the macroscopic field equation describing dynamic phenomena in disordered media can be expressed in which have been recently adopted in describing the terms of fractional operators, complementary phenomenon of anomalous diffusion in disordered systems [13,14]. The
M. GIONA
206
fractional calculus therefore appears to be an appropriate analytical tool for the description of correlations in dynamics on disordered structures. A first interesting application of the above results, and in particular of the more tractable time-independent wave equation (6), is given by the analysis of a boundary condition problem in the presence of a potential step at a junction between a disordered and a Euclidean structure. This situation is shown in Fig. 1. Let the energy of the particles E be less than the potential Vo(E < V,). In the case of Fig. 1, the Schrodinger equations read dr2
+ 2a dr * dr2
+ k;q
= 0, x < 0, (17)
+ k2q = 0, x > 0,
where k$ = 2mEb*, k* = 2m(E - V,)/h* and r = 1x1. Positing B = (Y+ ik+, k+ = [ki - a * ] ‘12, the solution of equation q = Al exp(@)
+ A2exp(B*r),
I/J = Br exp(ikx)
+ B,exp(-ikx),
where the star indicates the complex conjugate. The last equation must satisfy continuity conditions
(17) is given by
x < 0, x > 0,
at x = 0
A, + A2 = B, + B2, /_?A, + B*A2 = ik(B1 leading
- B2),
to B,
=
p-b* -B*
A
2
=
A
p-ik -p*
A +
B*+ik
+
’
+ ik ik
(19)
’
-p*
+
equal
ik
B2’
to zero.
Therefore,
the solution
(_~~+pTk)Alev(-=) =ia+yL: _JAlexP(--Kx). (20) ”
Fig. 1. Schematic
2’
2ik
+
Since V. > E, k = iK and B2 must be identically of the wave equation for x > 0 reads
v=
B
-/?I* + ik
representation
II--
..~_._..____.__
of disordered medium-Euclidean step barrier in the Euclidean
“,
system junction medium.
in the presence
of a potential
Quantum
mechanics
in disordered
Positing qQ= MA 1exp ( --Kx), T = (M12/4, z = V,/E,
sion coefficient,
systems
201
T can be regarded
as the transmis-
one readily obtains
T=(1 -
(1 - v,‘> ql,‘) + (rp -
qz -
(21)
1])2’
which should be compared with the Euclidean solution T = l/z. Given equation (21), if the potential step satisfies the condition V. = E(l + q2) (22) then T = 1. This peculiar effect, which implies a nonmonotonic behaviour of the transmission coefficient T, Fig. 2, is a consequence of disorder, i.e. of Q,f 0. It can be intuitively understood by observing that, due to its anomalous properties in diffusional and vibrational dynamics, a disordered medium acts as a potential barrier for a particle of energy E whose amplitude is E $. In this way, equation (22) can be regarded as the condition under which particles in the disordered medium ‘feel’ the entire x-axis as an equipotential. This phenomenon is intuitive if one observes that an exponent D, greater than 2 implies correlation in the random motion, which in the case of the Schrbdinger equation appears in the form of an equivalent extra potential energy of amplitude Eq2. Since A - p-l - E-@, implying that the relaxation length is shorter for more energetic particles, it is natural to expect the extra potential due to the presence of the relaxation length to be an increasing function of the energy E. The complementary case is shown in Fig. 3 for the transition from a Euclidean to a disordered system in which a potential step is present. As in the previous case, we have
d2w +k&+9=0, x dx2 &JJ dV +2&z+ dr2 which, after some algebraic manipulation, T=
coefficient
(23)
k;t,b=O,x>O,
leads to the following expression for T 1 (24)
1 + (q + V[z + QzJ2 - 11).
1 Fig. 2. Transmission
< 0,
T vs z for
2 the
z
case of Fig. 1: (a) (d) D, = 3.5.
3 D, = 2.0;
(b)
D, = 2.5;
(c)
D, = 3.0;
M. GIONA
208
V
Fig. 3. The complementary
case of that of Fig. 1, when a potential
barrier
is present
in the disordered
system
As in the case of Fig. 1, the disordered medium acts with respect to the particle in the Euclidean medium as an extra barrier of energy Eq?, which is the complementary result of equation (22). To sum up, the presence of anomalous behaviour in diffusion (Dw > 2) leads to highly nontrivial predictions for quantum particles in a disordered medium. These predictions can be unified by means of the hypothesis that temporal correlations in random motion are equivalent to an additional potential whose magnitude is proportional to the energy of the particles. The analysis presented is a first step towards an understanding of anomalous scaling in the dynamics of disordered systems by means of field equations. It seeks to stimulate the formulation of a consistent, even if approximate, theory of anomalous behaviour capable of furnishing predictions or of falsification by experimental results in the other branches of physics. In particular, attention should be drawn to the effect of disordered and fractal interfaces on the structure of surface energy levels, since this could lead to meaningful results in surface physics and in the theory of adsorption. It is important to point out the relationship existing between the present work and the contributions on fractal space-time and fractal ideas in quantum mechanics. It is known [15, 161, that the Hausdorff dimension of a quantum mechanical path (e.g. a quantum oscillator) is equal to 2. From this and from the theoretical arguments (see e.g. Nottale [17]) the idea of a fractal space-time was developed and related to the Heisenberg’s uncertainty principle 118-211. The present analysis does not seek to present new results on the fundamental problem of a Cantor-like model of space-time but rather to derive from scaling arguments developed for disordered fractal systems, some of their implications in terms of classical quantum mechanics. It is possible that some of these results may prove useful in the formulation of a unified theory of quantum mechanics in fractal spaces. However, it should be pointed out that the description of diffusion processes on fractals as well as the result expressed by equation (14) in terms of fractional derivative are the manifestation of the appearance of correlations in the random motion on fractals and are not related to the definition of a fractal time. Finally, it may be pointed out that the modification of the Schrodinger equation, as a result of scaling arguments, and the phenomenon of energy shifting are not surprising since they are ultimately a consequence of the non-Brownian nature (in the sense of usual Wiener processes) of the fluctuations in anomalous disordered media. Acknowledgements-The
author
is grateful
to H. E. Roman
for useful suggestions.
Quantum
mechanics
in disordered
systems
209
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11.
12. 13. 14.
15. 16. 17. 18. 19. 20. 21.
B. B. Mandelbrot, The Fractul Geometry of Nature. Freeman, San Francisco (1982). L. Pietronero and E. Tosatti (eds), Fructals in Physics. North Holland, Amsterdam (1985). A. Bunde and S. Havlin (eds), Fractul and Disordered Systems. Springer, Berlin (1991). P. G. de Gennes, La Recherche 7, 919 (1976). S. Havlin and D. Ben-Avraham, Diffusion in disordered media, Adv. Phys. 36, 695-798 (1987). S. Alexander and R. Orbach, J. Phys. Lett. 43, L625 (1982). R. Rammal and G. Toulouse, J. Whys. Len. 44, L13 (1983). H. E. Roman, S. Russ and A. Bunde, Pfrys. Rev. Len. 44, 1643 (1991). R. Orbach, Science 231, 814 (1986). For the numerical analysis of the probability density function for diffusion (P(r, t)) where r is a Cartesian radial coordinate see: H. E. Roman, A. Bunde and S. Havlin, in Reluxation in Disordered Systems and Related Topics, edited by A. Campbell and C. Giovannella. Plenum Press, New York, (1990); H. E. Roman, A. Bunde and S. Havlin, Ber. Bunsenges. Phys. Chem. 93, 1205 (1989). Theoretical contributions on the functional form of (P(r, t)) employing the Cartesian coordinate r have been made by: R. A. Gouyer, Phys. Rev. A 32, 2324 (1985) and by: B. O’Shaugnessy and I. Procaccia, Phys. Rev. Lett. 54, 455 (1985). K. Oldham and J. Spanier, The Fractional CulcuZus. Academic Press, New York (1974). M. Giona and H. E. Roman, A theory of transport phenomena in disordered media, Chem. Eng. J. 49, 1 (1992). M. Giona and H. E. Roman, Fractional diffusion equation on fractals part I, J. Phys. A. 25, 2093-2106 (1992); H. E. Roman and M. Giona, Fractional diffusion equation on fractals Part II, J. Phys. A 25, 2107-2118 (1992). L. F. Abbott and M. B. Wise, Am. J. Phys. 49, 37-39 (1981). E. Campesino-Romeo, J. C. D’Olivo and M. Socolovsky, Phys. Lett. 89A, 321-324 (1982). L. Nottale, Inc. J. Mod. Phys. 4, 5047-51 (1989). G. N. Ord, J. Phys. A 16, 1869-1884 (1983). M. S. El Naschie, Quantum mechanics and the possibility of a Cantorian space-time, Chaos, Solitons & Fructals 1, 485-488 (1991). M. S. El Naschie, On the uncertainty of information in quantum space-time, Chaos, Solitons & Fractuls 2, 91-94 (1992). M. S. El Naschie, Multi-dimensional Cantor sets in classical and quantum mechanics, Chaos, Solitons & Fructals 2, 211-220 (1992).