Quantization with Fractional Calculus - Semantic Scholar

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,


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


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


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


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


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


7