atd d. ,. (11) where, for each order of derivative in the. Lagrangian, the generalized coordinates. ( ) isr q. , are defined as. ( ). ( ). ( ). ( ). ( )is r is bisr isr btd xd q q. â.
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
Quantization with Fractional Calculus EQAB M. RABEI Science Department Jerash Private Universit Jerash Physics Department Mutah University Mutah- Karak JORDAN ABDUL-WALI AJLOUNI and HUMAM B. GHASSIB Physics Departmen University of Jordan Amman JORDAN Abstract: As a continuation of Riewe’s pioneering work [Phys. Rev. E 55, 3581(1997)], the canonical quantization with fractional drivatives is carried out according to the Dirac method. The canonical conjugate-momentum coordinates are defined and turned into operators that satisfy the commutation relations, corresponding to the Poisson-bracket relations of the classical theory. These are generalized and the equations of motion are redefined in terms of the generalized brackets. A generalized Heisenberg equation of motion containing fractional derivatives is introduced. Key-Words:- Hamiltonian Formulation, Canonical Quantization, Fractional Calculus, Nonconservative systems.
inelastic scattering, electrical resistance,
1. Introduction Most
advanced
methods
and many other processes in nature.
of with
Many attempts have been made
conservative systems, although all natural
to incorporate nonconservative forces into
processes in the physical world are
Lagrangian
nonconservative. Classically or quantum-
formulations; but those attempts could not
mechanically treated, macroscopically or
give a completely consistent physical
microscopically viewed, the physical
interpretation
world shows different kinds of dissipation
Rayleigh dissipation function, invoked
and irreversibility. Mostly ignored in
when the frictional force is proportional
analytical techniques, this dissipation
to the velocity [1], was the first to be used
appears in friction, Brownian motion,
to describe frictional forces in the
classical
mechanics
deal
only
Lagrangian.
1
and
of
Hamiltonian
these
However,
forces.
in
The
that case,
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
another scalar function was needed, in
construct
addition to the Lagrangian, to specify the
Hamiltonian for such systems [2, 3]. In
equations of motion. At the same time,
particular, he has shown that using
this function does not appear in the
fractional derivatives it is possible to
Hamiltonian. Accordingly, the whole
construct
process is of no use when it is attempted
description of nonconservative systems,
to quantize nonconservative systems.
including Lagrangian and Hamiltonian
The most substantive work in this
mechanics,
the
Lagrangian
a
complete
canonical
and
the
mechanical
transformations,
context was that of Riewe[2,3] who used
Hamilton-Jacobi theory, and quantum
fractional
study
mechanics. But the wave function for the
nonconservative systems and was able to
damped harmonic oscillator is written in
generalize the Lagrangian and other
terms of three coordinates x , x 1 , and
derivatives
to
2
classical functions to take into account
x− 1 ; while we have two canonical
nonconservative effects.
2
As a sequel to Riewe's work,
conjugate momenta. Thus, one of the
Rabei et al. [4] used Laplace transforms
coordinates is not physical. In addition,
of fractional integrals and fractional
Riewe has mentioned neither Poisson's
derivatives to develop a general formula
brackets nor the commutators. Riewe also
for the potential of any arbitrary force,
did not consider the causality so a mistake
conservative or nonconservative. This led
has appeared when he apply his theorem
directly to the consideration of the
on the example which he has introduced
dissipative effects in Lagrangian and
as an illustration[8, 9].
Hamiltonian formulations.
In this paper we will show how to
Nonconservative systems can be
quantize nonconservative, or dissipative,
incorporated easily into the equation of
systems using fractional calculus. The
motion using the Newtonian procedure;
correct canonical conjugate variables will
but it is difficult to quantize systems with
be determined. The Poisson brackets and
this procedure. The only scheme for the
the
quantization of dissipative systems seems
generalized
to
quantization
derivatives. Besides, the equation of
procedure [5]. This procedure leads to the
motion in terms of Poisson brackets will
nonlinear
Schrödinger-Langevin
be introduced, and the wave function for
equation. The reason for the impossibility
the damped harmonic oscillator will be
of
obtained in terms of two canonical
be
the
the
stochastic
direct
quantization
of
nonconservative systems is the absence of
quantum
commutators to
include
will
be
fractional
coordinates.
the proper Lagrangian or Hamiltonian.
The paper is arranged as follows.
Riewe has used fractional calculus to
In Section 2, we introduce some concepts
2
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
of fractional calculus. In Section 3,
where, for each order of derivative in the
Riewe's fractional Hamiltonian mechanics
Lagrangian,
is reviewed. In Section 4, the canonical
coordinates q
the r , s (i )
are defined as
conjugate variables are determined. This leads
to
Poisson’s
brackets,
the
q
generalized Hamilton’s equation in terms
r , s (i )
=q
r , s (i ),b
=
generalized
d
s (i )
xr
d (t − b )
s (i )
.
(12)
of these brackets, and the commutation
Here s (i ) can be any non-negative real
relations. In addition, we introduce a
number. We define s (0) to be 0; so that
generalized
form
q
equation
motion.
of
of
Heisenberg's An
illustrative
r ,s (0 )
denotes the coordinate
x r [2,
3].
is
In Eq.s (11) and (12) Riewe used left
discussed according to our quantization
hand differentiations on right handed
procedure in Section 5. Some concluding
coordinates, we think that this what
remarks follow in Section 6.
causes the mistake appeared in his
example,
given
by
Riewe[2,3],
illustration[2,3,8]. To go over this conflict
2.
Riewe’s
we will use the left handed coordinates,
Fractional
i.e.,
Hamiltonian Mechanics Riewe [2, 3] started with the
L(qr , s (i ) , t )
Lagrangian
which
is
q
a
r , s (i )
=q
r , s (i ), a
=
d
s (i )
xr
d (t − a )
s (i )
function of time t and the set of all qr , s (i ) ,
over the whole present work, in Riewe's
where r = 1, ⋅ ⋅⋅, R indicates the particular
an in ours. This will introduce the causal
coordinate
appearance of our work. If in any case the
(forexampl
Lagrangian is an anticausal or a mixed
x1 = x, x2 = y, x3 = z ) and s (i ) indicates the
order
of
i = 1, ⋅ ⋅⋅, N .
i th
the
He
then
one, then Riewe's original equations and
derivative, used
definitions may be used with a correction
the
of use the left operation with the left
conventional calculus of variations in
coordinates, and so the right operation
classical
with the right coordinates.
mechanics
following
to
generalized
obtain
the
Euler-Lagrange
In order to derive the generalized
equation:
Hamilton’s equations, Riewe [2, 3] defined the generalized momenta as
∑ (− 1) N
i =0
s (i )
d
s (i )
d (t − a )
s (i )
∂L = 0, ∂qr ,s ( i )
follows:
(11)
3
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
pr , s ( i ) = pr , s ( i ), a =
This canonical-conjugate relation
N − i −1
∑ (−1) s ( k + i +1) − s (i +1) k =0
d s ( k + i +1) − s ( i +1) d (t − a ) s ( k + i +1) − s (i +1)
could
be
obtained
directly
from
Hamilton’s equation defined by Riewe [2,
⎫⎪ ⎧⎪ ∂L ×⎨ ⎬ ⎪⎩ ∂qr , s ( k + i +1) ⎪⎭
3], Eq. (16), as follows:
∂H = qr , s ( i +1) ∂pr , s ( i )
(13) Thus, the Hamiltonian reads N
H = ∑ qr , s ( i ) pr , s ( i −1) − L ,
d s(i +1 )− s ( i ) ⎛ d s ( i ) − s(i +1 ) ⎞ ⎟qr , s(i +1 ) ⎜ d(t-a)s(i +1 )− s ( i ) ⎜⎝ d (t − a ) s ( i ) − s(i +1 ) ⎟⎠
=
(14)
i =1
and the Hamilton’s equations of motion are defined as [2, 3]
d s ( i +1) − s ( i ) ∂H pr , s ( i ) = (−1) s ( i +1) − s ( i ) d (t − a ) s (i +1) − s ( i ) ∂qr , s ( i ) (15)
=
d s(i +1 )− s ( i ) qr , s (i ) , d(t-a)s(i +1 )− s ( i )
.
0 ≤ i ≤ N-1 (18)
We conclude that pr , s ( i ) is the canonical
∂H = qr , s ( i +1) ; ∂pr , s (i )
(16)
conjugate of qr , s ( i ) . We can then introduce the
and
∂H ∂L =− . ∂t ∂t
Hamiltonian in the form (17)
d s ( i +1) − s ( i ) −L, H =∑ q p s ( i +1) − s ( i ) r , s ( i ) r , s ( i ) i = 0 d (t − a ) 0 ≤ i ≤ N-1 N −1
4.
Quantization
with
Fractional Calculus 4.1 Canonical Conjugate Variables
=
and Poisson Brackets
N −1
∑q i =0
The process of quantizing the
This
r , s ( i +1)
is
pr , s ( i ) − L .
equivalent
(19) to
Riewe’s
Hamiltonian starts with changing the
Hamiltonian. It is applicable to higher-
coordinates qr , s ( i ) and momenta pr , s ( i )
order
which
correspond
to
Now, let us define the most general classical Poisson bracket for any
theory [11]. But the first step in our work
two functions, F and G, in phase space:
is to determine which of the pr , s ( i ) and the
canonical
integral
Teixeira [12].
the
Poisson-bracket relations of the classical
qr , s (i ) are
with
derivatives obtained by Pimental and
into operators satisfying commutation relations
Lagrangians
conjugate
variables.
4
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
generalized definitions are applicable for N −1
{F , G} = ∑∑ r
k =0
∂F ∂G − ∂qr , s ( k ) ∂pr , s ( k )
∂F ∂G ∂pr , s ( k ) ∂q r , s ( k )
fractional and integral systems as well.
4.2 Quantum Mechanical Operator
.
Brackets
(20)
We
The fundamental Poisson brackets read N −1
∂qr , s (i ) ∂pl , s ( j )
m k =0
∂qm , s ( k ) ∂pm , s ( k )
{q
r , s ( i ) , pl , s ( j ) } = ∑∑
∂qr , s (i ) ∂pl , s ( j ) ∂pm , s ( k ) ∂q m, s ( k )
connect
the
mechanically by defining the momentum
−
operator as
ps ( i )
= δ ijδ rl .
now
canonical conjugate variables quantum
0 ≤ i, j ≤ N-1
,
can
(21)
The
=
η ∂ , i ∂qs (i )
i = 0,1,..., N − 1. (24)
correspondence
between
the
Substituting integral derivatives, one can
quantum-mechanical operator bracket and
recover the well-known definition of
the classical Poisson bracket is
Poisson brackets.
[qr , s (i ) , pr , s (i ) ]Ψ = [qr , s (i ) pr , s (i ) − pr , s (i ) qr , s (i ) ]Ψ
According to our definition of the
(25)
Hamiltonian, Hamilton’s equations of motion can be written in terms of Poisson brackets as
=
d s ( i +1) − s ( i ) qr , s ( i ) = qr , s (i +1) ={qr , s ( i ) , H } d (t − a ) s (i +1) − s ( i )
HΨ = iη
s ( i +1) − s ( i )
− {pr , s ( i ) , H }
= iηΨ ; (26) and the Schrödinger equation reads
(22)
(− 1)s (i +1) − s (i )
⎤ ∂ ∂ η⎡ − qs ( i ) ⎥ Ψ ⎢ qs ( i ) i ⎢⎣ ∂qs ( i ) ∂qs ( i ) ⎥⎦
d pr , s ( i ) = d (t − a) s ( i +1) − s ( i )
∂ Ψ . (27) ∂t
Thus, the commutators of the quantum-mechanical
(23)
proportional
These two definitions are valid for higher-
classical Poisson brackets:
order
Lagrangians
derivatives
and
definitions
given
lead by
with to
to
operators the
are
corresponding
integral the
Pimental
same
[qr , s (i ) , pr , s (i ) ] ↔ iη{qr , s (i ) , pr , s (i ) }. (28)
and
Teixeira [12]. This means that our
5
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
4.3
Generalization
of
leads to complicated nonlinear equations
Heisenberg's
such as Brownian motion.
Equation of Motion
References For any operator Q , Heisenberg's
[1] H. Goldstein, Classical Mechanics (2nd ed., Addison-Wesley, 1980). equation of motion states that [13, 14] [2] F. Riewe, Physical Review E 53,1890 (1996). d ˆ 1 ˆ ˆ Q= Q, H . (29) [3] F. Riewe, Physical Review E 55,3581 dt iη (1997). [4] ] E. M. Rabei, T. Al-halholy and A. This equation can be generalized for Rousan,., International Journal of coordinate operators as Modern Physics A, 19,3083, (2004). [5] K. Hajra, Stochastic Equation of a Dissipative Dynamical Systems and its 1 d s ( i +1) − s (i ) Hydrodynamical Interpretation, = qˆ qˆr , s ( i ) , Hˆ , s ( i +1) − s ( i ) r , s ( i ) Journal of Mathematical Physics. 32,1505 d (t − a) iη (1991). (30) [6] B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, and for momentum operators as NewYork, 1974). [7] A. Carpintri and F. Mainardi, Fractals and Fractional Calculus in Continuum 1 d s (i +1) − s ( i ) s ( i +1) − s ( i ) (−1) pˆ r , s ( i ) = − pˆ r , s (i ) , Hˆ Mechanics (Springer , New York, s ( i +1) − s ( i ) 1997). d (t − a) iη [8] D.W. Dreisigmeyer and M. Young , . (31) Journal of Physics A: Mathematical and Equations (30) and (31) are valid General 36, 8279 (2003).
[ ]
[
]
[
]
[9] D. W. Dreisigmeyer, and M. Young, http://www.arXiv:physics/0312085 v1(2003).
for integer-order derivatives as well as non-integer order.
[10] O.P. Agrawal, Journal of Mathematical Analysis and Applications 272, 368 (2002). [11] P. A. M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964). [12] B. M. Pimental and R. G. Teixeira, arXiv:hep-/9704088 v1 (1997). [13] P.T.Matthews, Introduction to Quantum Mechanics ( 2nd ed., McGrawHill Ltd., London, 1974). [14] J. J. Sakurai, Modern Quantum Mechanics (Benjamin/Cummings, Menlo Park, CA, 1985). [15] D. J. Griffiths, Introduction to Quantum Mechanics (Prentice-Hall, New Jersey, 1995). [16] E. Merzbacher, Quantum Mechanics (Wiley, NewYork, 11970)
6. Conclusion We have demonstrated that the canonical quantization procedure can be applied to nonconservative systems using fractional derivatives. This procedure should be very helpful in quantizing nonconservative systems
related
to
many
important
physical problems: either where the ordinary quantum-mechanical treatment leads to an incomplete description, such as the energy loss by charged particles when passing through matter; or where it
6
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
7