Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis
© Yang et al.; licensee Springer 2013
Received: 29 January 2013
Accepted: 2 May 2013
Published: 20 May 2013
In this paper, we discuss the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. The Schrödinger equation and Heisenberg uncertainty principles are structured within local fractional operators.
KeywordsHeisenberg uncertainty principle local fractional Fourier operator Schrödinger equation fractal time-space
As it is known, the fractal curves [1, 2] are everywhere continuous but nowhere differentiable; therefore, we cannot use the classical calculus to describe the motions in Cantor time-space [3–10]. The theory of local fractional calculus [11–20], started to be considered as one of the useful tools to handle the fractal and continuously non-differentiable functions. This formalism was applied in describing physical phenomena such as continuum mechanics , elasticity [20–22], quantum mechanics [23, 24], heat-diffusion and wave phenomena [25–30], and other branches of applied mathematics [31–33] and nonlinear dynamics [34, 35].
The fractional Heisenberg uncertainty principle and the fractional Schrödinger equation based on fractional Fourier analysis were proposed [36–48]. Local fractional Fourier analysis , which is a generalization of the Fourier analysis in fractal space, has played an important role in handling non-differentiable functions. The theory of local fractional Fourier analysis is structured in a generalized Hilbert space (fractal space), and some results were obtained [26, 49–53]. Also, its applications were investigated in quantum mechanics , differentials equations [26, 28] and signals .
The main purpose of this paper is to present the mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis and to structure a local fractional version of the Schrödinger equation.
The manuscript is structured as follows. In Section 2, the preliminary results for the local fractional calculus are investigated. The theory of local fractional Fourier analysis is introduced in Section 3. The Heisenberg uncertainty principle in local fractional Fourier analysis is studied in Section 4. Application of quantum mechanics in fractal space is considered in Section 5. Finally, the conclusions are presented in Section 6.
2 Mathematical tools
2.1 Local fractional continuity of functions
2.2 Local fractional derivative and integration
where with a gamma function .
where , and , , , , is a partition of the interval .
3 Theory of local fractional Fourier analysis
In this section, we investigate local fractional Fourier analysis [49–53], which is a generalized Fourier analysis in fractal space. Here we discuss the local fractional Fourier series, the Fourier transform and the generalized Fourier transform in fractal space. We start with a local fractional Fourier series.
3.1 Local fractional Fourier series
The above is generalized to calculate the local fractional Fourier series.
3.2 The Fourier transform in fractal space
where the latter converges.
3.3 The generalized Fourier transform in fractal space
where with .
where with .
3.4 Some useful results
4 Heisenberg uncertainty principles in local fractional Fourier analysis
with and a constant .
When , then we have with a constant .
when , .
Hence, this result is obtained. □
As a direct result, we have two equivalent forms as follows.
with equality only if is almost everywhere equal to a constant multiple of , with and a constant .
Hence, Theorem 5 is obtained. □
5 The mathematical aspect of fractal quantum mechanics
5.1 Local fractional Schrödinger equation
where with .
where is the local fractional Hamiltonian in fractal mechanics.
where is non-differential action, is the local fractional Hamiltonian function, and () are generalized fractal coordinates.
5.2 Solutions of the local fractional Schrödinger equation
5.2.1 General solutions of the local fractional Schrödinger equation
5.2.2 Fractal complex wave functions
5.2.3 Probabilistic interpretation of fractal complex wave function of one variable
5.3 The Heisenberg uncertainty principle in fractal quantum mechanics
where h is Planck’s constant.
In this manuscript, the uncertainty principle in local fractional Fourier analysis is suggested. Since the local fractional calculus can be applied to deal with the non-differentiable functions defined on any fractional space, the local fractional Fourier transform is important to deal with fractal signal functions. The results on uncertainty principles could play an important role in time-frequency analysis in fractal space. From Eq. (A.7) we conclude that there is a semi-group property for the Mittag-Leffler function on fractal sets. Meanwhile, uncertainty principles derived from local fractional Fourier analysis are classical uncertainty principles in the case of . We reported the structure the local fractional Schrödinger equation derived from Planck-Einstein and de Broglie relations in fractal time space.
Dedicated to Professor Hari M Srivastava.
The authors would like to thank to the referees for their very useful comments and remarks.
- Mandelbrot BB: The Fractal Geometry of Nature. Freeman, New York; 1982.MATHGoogle Scholar
- Falconer KJ: Fractal Geometry-Mathematical Foundations and Application. Wiley, New York; 1997.MATHGoogle Scholar
- Zeilinger A, Svozil K: Measuring the dimension of space-time. Phys. Rev. Lett. 1985, 54(24):2553–2555. 10.1103/PhysRevLett.54.2553View ArticleGoogle Scholar
- Nottale L: Fractals and the quantum theory of space-time. Int. J. Mod. Phys. A 1989, 4(19):5047–5117. 10.1142/S0217751X89002156MathSciNetView ArticleGoogle Scholar
- Saleh AA: On the dimension of micro space-time. Chaos Solitons Fractals 1996, 7(6):873–875. 10.1016/0960-0779(96)00022-7MathSciNetView ArticleGoogle Scholar
- Maziashvili, M: Space-time uncertainty relation and operational definition of dimension. (2007) arXiv:0709.0898MATHGoogle Scholar
- Caruso F, Oguri V: The cosmic microwave background spectrum and an upper limit for fractal space dimensionality. Astrophys. J. 2009, 694(1):151–156. 10.1088/0004-637X/694/1/151View ArticleGoogle Scholar
- Calcagni G: Geometry and field theory in multi-fractional spacetime. J. High Energy Phys. 2012, 65(1):1–65.MathSciNetMATHGoogle Scholar
- Kong HY, He JH: A novel friction law. Therm. Sci. 2012, 16(5):1529–1533. 10.2298/TSCI1205529KView ArticleGoogle Scholar
- Kong HY, He JH: The fractal harmonic law and its application to swimming suit. Therm. Sci. 2012, 16(5):1467–1471. 10.2298/TSCI1205467KView ArticleGoogle Scholar
- Kolwankar KM, Gangal AD: Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 1996, 6(4):505–513. 10.1063/1.166197MathSciNetView ArticleMATHGoogle Scholar
- Jumarie G: Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Appl. Math. Lett. 2009, 22: 378–385. 10.1016/j.aml.2008.06.003MathSciNetView ArticleMATHGoogle Scholar
- Parvate A, Gangal AD: Calculus on fractal subsets of real line - I: formulation. Fractals 2009, 17(1):53–81. 10.1142/S0218348X09004181MathSciNetView ArticleMATHGoogle Scholar
- Chen W: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals 2006, 28: 923–929. 10.1016/j.chaos.2005.08.199View ArticleMATHGoogle Scholar
- Adda FB, Cresson J: About non-differentiable functions. J. Math. Anal. Appl. 2001, 263: 721–737. 10.1006/jmaa.2001.7656MathSciNetView ArticleMATHGoogle Scholar
- Balankin AS, Elizarraraz BE: Map of fluid flow in fractal porous medium into fractal continuum flow. Phys. Rev. E 2012., 85(5): Article ID 056314Google Scholar
- He JH: A new fractal derivation. Therm. Sci. 2011, 15: 145–147.Google Scholar
- Yang XJ: Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4: 1–225.Google Scholar
- Yang XJ: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher, Hong Kong; 2011.Google Scholar
- Yang XJ: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York; 2012.Google Scholar
- Carpinteri A, Chiaia B, Cornetti P: Static-kinematic duality and the principle of virtual work in the mechanics of fractal media. Comput. Methods Appl. Mech. Eng. 2001, 191: 3–19. 10.1016/S0045-7825(01)00241-9View ArticleMATHGoogle Scholar
- Carpinteri A, Cornetti P: A fractional calculus approach to the description of stress and strain localization in fractal media. Chaos Solitons Fractals 2002, 13(1):85–94. 10.1016/S0960-0779(00)00238-1View ArticleMATHGoogle Scholar
- Yang XJ: The zero-mass renormalization group differential equations and limit cycles in non-smooth initial value problems. Prespacetime J. 2012, 3(9):913–923.Google Scholar
- Kolwankar KM, Gangal AD: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 1998, 80: 214–217. 10.1103/PhysRevLett.80.214MathSciNetView ArticleMATHGoogle Scholar
- Wu GC, Wu KT: Variational approach for fractional diffusion-wave equations on Cantor sets. Chin. Phys. Lett. 2012., 29(6): Article ID 060505Google Scholar
- Zhong WP, Gao F, Shen XM: Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral. Adv. Mater. Res. 2012, 461: 306–310.View ArticleGoogle Scholar
- Yang XJ, Baleanu D: Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 2012. doi:10.2298/TSCI121124216YGoogle Scholar
- Hu MS, Agarwal RP, Yang XJ: Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstr. Appl. Anal. 2012., 2012: Article ID 567401Google Scholar
- Yang, XJ, Baleanu, D, Zhong, WP: Approximation solution to diffusion equation on Cantor time-space. Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. (2013, in press)
- Hu MS, Baleanu D, Yang XJ: One-phase problems for discontinuous heat transfer in fractal media. Math. Probl. Eng. 2013., 2013: Article ID 358473Google Scholar
- Babakhani A, Gejji VD: On calculus of local fractional derivatives. J. Math. Anal. Appl. 2002, 270(1):66–79. 10.1016/S0022-247X(02)00048-3MathSciNetView ArticleMATHGoogle Scholar
- Chen Y, Yan Y, Zhang K: On the local fractional derivative. J. Math. Anal. Appl. 2010, 362: 17–33. 10.1016/j.jmaa.2009.08.014MathSciNetView ArticleMATHGoogle Scholar
- Kim TS: Differentiability of fractal curves. Commun. Korean Math. Soc. 2005, 20(4):827–835.MathSciNetView ArticleMATHGoogle Scholar
- Parvate A, Gangal AD: Fractal differential equations and fractal-time dynamical systems. Pramana J. Phys. 2005, 64(3):389–409. 10.1007/BF02704566View ArticleGoogle Scholar
- Yang XJ, Liao MK, Wang JN: A novel approach to processing fractal dynamical systems using the Yang-Fourier transforms. Adv. Electr. Eng. Syst. 2012, 1(3):135–139.Google Scholar
- Namias V: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 1980, 25(3):241–265. 10.1093/imamat/25.3.241MathSciNetView ArticleMATHGoogle Scholar
- Mustard D: Uncertainty principles invariant under the fractional Fourier transform. J. Aust. Math. Soc. 1991, 33(2):180–191. 10.1017/S0334270000006986MathSciNetView ArticleMATHGoogle Scholar
- Bhatti M: Fractional Schrödinger wave equation and fractional uncertainty principle. Int. J. Contemp. Math. Sci. 2007, 19(2):943–950.MathSciNetMATHGoogle Scholar
- Laskin N: Fractional quantum mechanics. Phys. Rev. E 2000, 62: 3135–3145. 10.1103/PhysRevE.62.3135View ArticleMATHGoogle Scholar
- Laskin N: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 2000, 268: 298–305. 10.1016/S0375-9601(00)00201-2MathSciNetView ArticleMATHGoogle Scholar
- Laskin N: Fractional Schrödinger equation. Phys. Rev. E 2002., 66: Article ID 056108Google Scholar
- Muslih SI, Agrawal OP, Baleanu D: A fractional Schrödinger equation and its solution. Int. J. Theor. Phys. 2010, 49(8):1746–1752. 10.1007/s10773-010-0354-xMathSciNetView ArticleMATHGoogle Scholar
- Adda FB, Cresson J: Quantum derivatives and the Schrödinger equation. Chaos Solitons Fractals 2004, 19: 1323–1334. 10.1016/S0960-0779(03)00339-4MathSciNetView ArticleMATHGoogle Scholar
- Tofighi A: Probability structure of time fractional Schrödinger equation. Acta Phys. Pol. 2009, 116(2):114–118.View ArticleGoogle Scholar
- Naber M: Time fractional Schrödinger equation. J. Math. Phys. 2004, 45(8):3325–3339.MathSciNetView ArticleMATHGoogle Scholar
- Dong JP, Xu MY: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 2007., 48: Article ID 072105Google Scholar
- Rozmej P, Bandrowski B: On fractional Schrödinger equation. Comput. Methods Sci. Technol. 2010, 16(2):191–194.MathSciNetView ArticleGoogle Scholar
- Iomin A: Fractional-time Schrödinger equation: fractional dynamics on a comb. Chaos Solitons Fractals 2011, 44: 348–352. 10.1016/j.chaos.2011.03.005MathSciNetView ArticleMATHGoogle Scholar
- Liao MK, Yang XJ, Yan Q: A new viewpoint to Fourier analysis in fractal space. In Advances in Applied Mathematics and Approximation Theory. Edited by: Anastassiou GA, Duman O. Springer, New York; 2013:399–411. Chapter 26Google Scholar
- Guo Y: Local fractional Z transform in fractal space. Adv. Digit. Multimed. 2012, 1(2):96–102.Google Scholar
- Yang XJ, Liao MK, Chen JW: A novel approach to processing fractal signals using the Yang-Fourier transforms. Proc. Eng. 2012, 29: 2950–2954.View ArticleGoogle Scholar
- Yang XJ: Theory and applications of local fractional Fourier analysis. Adv. Mech. Eng. Appl. 2012, 1(4):70–85.Google Scholar
- He JH: Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal. 2012., 2012: Article ID 916793Google Scholar
- He JH: Frontier of modern textile engineering and short remarks on some topics in physics. Int. J. Nonlinear Sci. Numer. Simul. 2010, 11(7):555–563.View ArticleGoogle Scholar
- Yang CD: Trajectory interpretation of the uncertainty principle in 1D systems using complex Bohmian mechanics. Phys. Lett. A 2008, 372(41):6240–6253. 10.1016/j.physleta.2008.08.050MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.