Application of Scale Relativity (ScR) Theory to the Problem of a Particle in a Finite One-Dimensional Square Well (FODSW) Potential

Saeed N. T Al-Rashid, Mohammed A. Z Habeeb, Khalid A Ahmad

.
Physics Department, College of Science, Al-Anbar University, Anbar, Iraq,.

DOI: 10.4236/jqis.2011.11002                

Abstract

In the present work, and along the lines of Hermann, ScR theory is applied to a finite one-dimensional square well potential problem. The aim is to show that scale relativity theory can reproduce quantum mechanical results without employing the Schrödinger equation. Some mathematical difficulties that arise when obtaining the solution to this problem were overcome by utilizing a novel mathematical connection between ScR theory and the well-known Riccati equation. Computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle in the above potential utilizing the solutions of the fractal equations of motion obtained from ScR theory. Several attempts were made to fix some of the parameters in the numerical simulations to obtain the best possible results in a practical computer CPU time within limited local computer facilities [1,2]. Comparison of the present results with the corresponding results obtained from conventional quantum mechanics by solving the Schrödinger equation, shows very good agreement. This agreement was improved further by optimizing the parameters used in the numerical simulations [1,3]. This represents a new example where scale relativity theory, based on a fractal space-time concept, can accurately reproduce quantum mechanical results without invoking the Schrödinger equation.

Keywords

Square Well, ScR Theory, Numerical Simulations, Fractal Space-Time

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	L. Nottale, “The Theory of Scale Relativity,” International Journal of Modern Physics A, Vol. 7, No. 20, 1992, pp. 4899-4936. doi:10.1142/S0217751X92002222
[2]	L. Nottale, “Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity,” World Scientific, Singapore, 1998.
[3]	L. Nottale, “The Scale Relativity Program,” Chaos, Solitons and Fractals, Vol. 10, No. 2-3, 1994, pp. 459-468.
[4]	doi:10.1016/S0960-0779(98)00195-7
[5]	L. Nottale, “Scale Relativity and Fractal Space-Time: Application to Quantum Physics, Cosmology and Chaotic Systems,” Chaos, Solitons and Fractals, Vol. 7, No. 6, 1996, pp. 877-938. doi:10.1016/0960-0779(96)00002-1
[6]	L. Nottale, “Scale Relativity, Fractal Space-Time and Quantum Mechanics,” Chaos, Solitons and Fractals, Vol. 4, No. 3, 1994, pp. 361-388.
[7]	doi:10.1016/0960-0779(94)90051-5
[8]	L. Nottale, “Scale Relativity, Fractal Space-Time and Morphogenesis of Structures”, Sciences of the Interface, Proceedings of Interna-tional Symposium in Honor of O. Rossler, ZKM Karlsruhe, 2000, p. 38.
[9]	L. Nottale, “Scale Relativity,” Reprinted from “Scale Invariance and Beyond,” In: B. Dubralle, F. Graner and D. Sornette, Eds., Proceedings of Les Houches, EDP Science, 1997, pp. 249-261.
[10]	L. Nottale, “Scale Relativity and Quantization of the Universe-I, Theoretical Framework,” As-tronomy and Astrophysics, Vol. 327, No. 3, 1997, pp. 867-889.
[11]	L. Nottale, G. Schumacher and J. Gray, “Scale Relativity and Quantization of the Solar System,” Astronomy and Astrophysics, Vol. 322, No. 3, 1997, pp. 1018-1025.
[12]	L. Nottale and M. N. Célérier, “Derivation of the Postulates of Quantum Mechanics from the First Principles of Scale Relativity,” Journal of Physics A: Mathematical and Theoretical, 2007, Vol. 40, No. 48, pp. 14471-14498.
[13]	doi:10.1088/1751-8113/40/48/012
[14]	M.-N. Célérier and L. Nottale, “Electromagnetic Klein-Gordon and Dirac equations in scale relativity,” International Journal of Modern Physics A, Vol. 25, No. 22, 2010, pp. 4239-4253.
[15]	L. Nottale, “On the Transition from the Clas-sical to the Quantum Regime in Fractal Space-Time Theory,” Chaos, Solitons and Fractals, 2005, Vol. 25, No. 4, pp. 797-803.
[16]	doi:10.1016/j.chaos.2004.11.071
[17]	R. P. Hermann, “Numerical Simulation of a Quantum Particle in a Box,” Journal of Physics A: Mathematical and General, Vol. 30, No. 11, 1997, pp. 3967-3975.
[18]	doi:10.1088/0305-4470/30/11/023
[19]	L. I. Schiff., “Quantum Mechanics,” 3rd Edition, Int. Student, McGraw-Hill, New York, 1969.
[20]	S. Gasiorowicz, “Quan-tum Physics,” John Wiley and Sons, New York, 1974.
[21]	J. L. Powell and B. Crasemann, “Quantum Mechan-ics,”
[22]	Addison-Wesley Co., Inc., Massachusetts, 1961.
[23]	C. C. Tannoudji, B. Diue and F. Laloё, “Quantum me-
[24]	chanics,” John Wiley and Sons, New York, 1977.
[25]	W. T. Reid, “Riccati Differential Equations,” Aca-demic Press, New York, 1972.
[26]	F. Charlton, “Integrating Factor for First-Order Differential Equations,” Classroom Notes, Aston University, Birmingham, 1998.
[27]	N. Bessis and G. Bessis, “Open Perturbation and Riccati Equation: Alge-braic Determination of Quartic Anharmonic Oscillator Energies and Eigenfunctions,” Journal of Mathematical Physics, Vol. 38, No. 11, 1997, pp. 5483-5492. doi:10.1063/1.532147
[28]	G. W. Rogers, “Riccati Equation and Perturbation Expansion in Quantum Mechanics,” Journal of Mathematical Physics, Vol. 26, No. 14, 1985, pp. 567-575.
[29]	doi:10.1063/1.526592
[30]	R. H. Dicke and J. P. Wittke, “Introduction to Quantum Mechanics,” Addi-son-Wesley Co., Inc., Massachusetts, 1960.
[31]	D. Po?ani?, “Bound State in a One-dimensional Square Potential Well in Quantum Mechanics,” Classroom Notes, University of Virginia, Charlottesville, 2002.