1 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, Non- conservative systems. 1. Introduction Most advanced methods of classical mechanics deal only with conservative systems, although all natural processes in the physical world are nonconservative. Classically or quantum- mechanically treated, macroscopically or microscopically viewed, the physical world shows different kinds of dissipation and irreversibility. Mostly ignored in analytical techniques, this dissipation appears in friction, Brownian motion, inelastic scattering, electrical resistance, and many other processes in nature. Many attempts have been made to incorporate nonconservative forces into Lagrangian and Hamiltonian formulations; but those attempts could not give a completely consistent physical interpretation of these forces. The Rayleigh dissipation function, invoked when the frictional force is proportional to the velocity [1], was the first to be used to describe frictional forces in the Lagrangian. However, in that case, Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
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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
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
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4.3 Generalization of Heisenberg's
Equation of Motion
For any operator Q , Heisenberg's
equation of motion states that [13, 14]
[ ]HQi
Qdtd ˆ,ˆ1ˆ
η= . (29)
This equation can be generalized for
coordinate operators as
[ ],ˆ,ˆ1ˆ)( )(,)(,)()1(
)()1(
Hqi
qatd
disrisrisis
isis
η=
− −+
−+
(30)
and for momentum operators as
[ ]Hpi
patd
disrisrisis
isisisis ˆ,ˆ1ˆ
)()1( )(,)(,)()1(
)()1()()1(
η−=
−− −+
−+−+
. (31)
Equations (30) and (31) are valid
for integer-order derivatives as well as
non-integer order.
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
leads to complicated nonlinear equations
such as Brownian motion.
References
[1] H. Goldstein, Classical Mechanics (2nd ed., Addison-Wesley, 1980). [2] F. Riewe, Physical Review E 53,1890 (1996). [3] F. Riewe, Physical Review E 55,3581 (1997). [4] ] E. M. Rabei, T. Al-halholy and A. Rousan,., International Journal of Modern Physics A, 19,3083, (2004). [5] K. Hajra, Stochastic Equation of a Dissipative Dynamical Systems and its Hydrodynamical Interpretation, Journal of Mathematical Physics. 32,1505 (1991). [6] B. Oldham and J. Spanier, The Fractional Calculus (Academic Press, NewYork, 1974). [7] A. Carpintri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics (Springer , New York, 1997). [8] D.W. Dreisigmeyer and M. Young , Journal of Physics A: Mathematical and General 36, 8279 (2003). [9] D. W. Dreisigmeyer, and M. Young, http://www.arXiv:physics/0312085 v1(2003). [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., McGraw- Hill 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)
Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)
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Proceedings of the 9th WSEAS International Conference on Applied Mathematics, Istanbul, Turkey, May 27-29, 2006 (pp256-262)