Abstract
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.
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1 Introduction
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 [21], 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 [49], 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 [23], differentials equations [26, 28] and signals [51].
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
If there is
with , for and . Now is called a local fractional continuous at , denoted by . Then is called local fractional continuous on the interval , denoted by
The function is said to be local fractional continuous at from the right if is defined, and
The function is said to be local fractional continuous at from the left if is defined, and
Suppose that , and , then we have the following relation:
For other results of theory of local fractional continuity of functions, see [18–20, 27–30].
2.2 Local fractional derivative and integration
Setting , a local fractional derivative of of order α at is defined by
where with a gamma function .
Setting, a local fractional integral of of order α in the interval is defined as
where , and , , , , is a partition of the interval .
Their fractal geometrical explanation of local fractional derivative and integration can be seen in [22, 26, 50–52].
with .
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 local fractional trigonometric Fourier series of is given by
Then the local fractional Fourier coefficients can be computed by
The Mittag-Leffler functions expression of the local fractional Fourier series is described by [18, 19, 49–52]
where the local fractional Fourier coefficients are
The above is generalized to calculate the local fractional Fourier series.
3.2 The Fourier transform in fractal space
Suppose that , the Fourier transform in fractal space, denoted by , is written in the form
where the latter converges.
If , its inversion formula is written in the form
3.3 The generalized Fourier transform in fractal space
The generalized Fourier transform in fractal space is written in the form
where with .
The inverse formula of the generalized Fourier transform in fractal space is written in the form [18, 19]
where with .
3.4 Some useful results
The following formula is valid [18, 19].
If , then we have
If , then we have
4 Heisenberg uncertainty principles in local fractional Fourier analysis
Theorem 4 Suppose that , , then we have
with equality only if is almost everywhere equal to a constant multiple of
with and a constant .
Proof Considering the equality
we have
When , then we have with a constant .
Since
and
we have
when , .
Hence, there is
such that
Therefore, we deduce to
Hence, this result is obtained. □
As a direct result, we have two equivalent forms as follows.
Theorem 5 Suppose that and , then we have
with equality only if is almost everywhere equal to a constant multiple of , with and a constant .
Proof Applying Theorem 4, we have
such that
Hence, Theorem 5 is obtained. □
The above results [37, 38] are different from the results in fractional Fourier transform [36, 37] based on the fractional calculus theory.
5 The mathematical aspect of fractal quantum mechanics
5.1 Local fractional Schrödinger equation
We structure the non-differential phase of a fractal plane wave as a complex phase factor using the relations
where the Planck-Einstein and De Broglie relations are in fractal space
We can realize the local fractional partial derivative with respect to fractal space
and fractal time
From (5.3) we have
such that
where with .
From (5.4) we have
We have the energy equation
such that
and
where is the local fractional Hamiltonian in fractal mechanics.
Hence, we have that
Therefore, we can deduce that the local fractional energy operator is
and that the local fractional momentum operator is
Therefore, we get the local fractional Schrödinger equation in the form of local fractional energy and momentum operators
where the local fractional Hamiltonian is
We also deduce that the general time-independent local fractional Schrödinger equation is written in the form
which is related to the following equation:
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
The general solution of the local fractional Schrödinger equation can be seen in the following. For discrete k, the sum is a superposition of fractal plane waves:
and
If we consider and , we have fractal plane waves:
5.2.2 Fractal complex wave functions
The meaning of this description can be seen in the following. Similar to the classical wave mechanics, we prepare N atoms independently, in the same state, so that when each of them is measured, they are described by the same wave function. Then the result of a position measurement is described as the fractal probability density, and we wish it is not the same for all. The set of impacts is distributed in space with the probability density
In view of (5.20), we have
The set of N measurements is characterized by an expectation value and a root mean square dispersion ,
Similarly, the square of the dispersion is defined by
If the physical interpretation of a particle in fractal space is that the probability
the integral of this quantity over all fractal space is
For (5.18) we have
such that
5.2.3 Probabilistic interpretation of fractal complex wave function of one variable
In (5.22), we have
and
such that
where
We have an expectation value and a root mean square dispersion ,
and
For a given fractal mechanical operatorA, we have an expectation value and a root mean square dispersion ,
and
5.3 The Heisenberg uncertainty principle in fractal quantum mechanics
Suppose that
we have a fractal positional operator expectation value
and a root mean square dispersion of positional operator
Similar to the fractal positional operator, we have a fractal momentum operator expectation value
and a root mean square dispersion of positional operator
Considering
and
by using Theorem 5, we have that
such that
Hence, we have that
such that
where
and
Suppose that
then we have
and
where [24].
The above equation (5.50) differs from the results presented in [36, 37]. Also, Eq. (5.51) is different from the ones reported in [38–40, 54, 55].
Below we define the local fractional energy operator
and the local fractional momentum operator
where [26].
Thus, we get the Planck-Einstein and de Broglie relations are in fractal space as
where h is Planck’s constant.
6 Conclusions
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.
Appendix
such that
where is a fractal integral staircase function. We have the relation [18–20]
such that
Inversely we obtain
Hence, both and are seen in [20, 21].
In view of Eq. (A.4), we easily obtain that
and
where , , and is a fractal unit of imaginary number [18–20, 53].
References
Mandelbrot BB: The Fractal Geometry of Nature. Freeman, New York; 1982.
Falconer KJ: Fractal Geometry-Mathematical Foundations and Application. Wiley, New York; 1997.
Zeilinger A, Svozil K: Measuring the dimension of space-time. Phys. Rev. Lett. 1985, 54(24):2553–2555. 10.1103/PhysRevLett.54.2553
Nottale L: Fractals and the quantum theory of space-time. Int. J. Mod. Phys. A 1989, 4(19):5047–5117. 10.1142/S0217751X89002156
Saleh AA: On the dimension of micro space-time. Chaos Solitons Fractals 1996, 7(6):873–875. 10.1016/0960-0779(96)00022-7
Maziashvili, M: Space-time uncertainty relation and operational definition of dimension. (2007) arXiv:0709.0898
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/694/1/151
Calcagni G: Geometry and field theory in multi-fractional spacetime. J. High Energy Phys. 2012, 65(1):1–65.
Kong HY, He JH: A novel friction law. Therm. Sci. 2012, 16(5):1529–1533. 10.2298/TSCI1205529K
Kong HY, He JH: The fractal harmonic law and its application to swimming suit. Therm. Sci. 2012, 16(5):1467–1471. 10.2298/TSCI1205467K
Kolwankar KM, Gangal AD: Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 1996, 6(4):505–513. 10.1063/1.166197
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.2008.06.003
Parvate A, Gangal AD: Calculus on fractal subsets of real line - I: formulation. Fractals 2009, 17(1):53–81. 10.1142/S0218348X09004181
Chen W: Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals 2006, 28: 923–929. 10.1016/j.chaos.2005.08.199
Adda FB, Cresson J: About non-differentiable functions. J. Math. Anal. Appl. 2001, 263: 721–737. 10.1006/jmaa.2001.7656
Balankin AS, Elizarraraz BE: Map of fluid flow in fractal porous medium into fractal continuum flow. Phys. Rev. E 2012., 85(5): Article ID 056314
He JH: A new fractal derivation. Therm. Sci. 2011, 15: 145–147.
Yang XJ: Local fractional integral transforms. Prog. Nonlinear Sci. 2011, 4: 1–225.
Yang XJ: Local Fractional Functional Analysis and Its Applications. Asian Academic Publisher, Hong Kong; 2011.
Yang XJ: Advanced Local Fractional Calculus and Its Applications. World Science Publisher, New York; 2012.
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)00241-9
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)00238-1
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.
Kolwankar KM, Gangal AD: Local fractional Fokker-Planck equation. Phys. Rev. Lett. 1998, 80: 214–217. 10.1103/PhysRevLett.80.214
Wu GC, Wu KT: Variational approach for fractional diffusion-wave equations on Cantor sets. Chin. Phys. Lett. 2012., 29(6): Article ID 060505
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.
Yang XJ, Baleanu D: Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci. 2012. doi:10.2298/TSCI121124216Y
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 567401
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 358473
Babakhani A, Gejji VD: On calculus of local fractional derivatives. J. Math. Anal. Appl. 2002, 270(1):66–79. 10.1016/S0022-247X(02)00048-3
Chen Y, Yan Y, Zhang K: On the local fractional derivative. J. Math. Anal. Appl. 2010, 362: 17–33. 10.1016/j.jmaa.2009.08.014
Kim TS: Differentiability of fractal curves. Commun. Korean Math. Soc. 2005, 20(4):827–835.
Parvate A, Gangal AD: Fractal differential equations and fractal-time dynamical systems. Pramana J. Phys. 2005, 64(3):389–409. 10.1007/BF02704566
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.
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/25.3.241
Mustard D: Uncertainty principles invariant under the fractional Fourier transform. J. Aust. Math. Soc. 1991, 33(2):180–191. 10.1017/S0334270000006986
Bhatti M: Fractional Schrödinger wave equation and fractional uncertainty principle. Int. J. Contemp. Math. Sci. 2007, 19(2):943–950.
Laskin N: Fractional quantum mechanics. Phys. Rev. E 2000, 62: 3135–3145. 10.1103/PhysRevE.62.3135
Laskin N: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 2000, 268: 298–305. 10.1016/S0375-9601(00)00201-2
Laskin N: Fractional Schrödinger equation. Phys. Rev. E 2002., 66: Article ID 056108
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-x
Adda FB, Cresson J: Quantum derivatives and the Schrödinger equation. Chaos Solitons Fractals 2004, 19: 1323–1334. 10.1016/S0960-0779(03)00339-4
Tofighi A: Probability structure of time fractional Schrödinger equation. Acta Phys. Pol. 2009, 116(2):114–118.
Naber M: Time fractional Schrödinger equation. J. Math. Phys. 2004, 45(8):3325–3339.
Dong JP, Xu MY: Some solutions to the space fractional Schrödinger equation using momentum representation method. J. Math. Phys. 2007., 48: Article ID 072105
Rozmej P, Bandrowski B: On fractional Schrödinger equation. Comput. Methods Sci. Technol. 2010, 16(2):191–194.
Iomin A: Fractional-time Schrödinger equation: fractional dynamics on a comb. Chaos Solitons Fractals 2011, 44: 348–352. 10.1016/j.chaos.2011.03.005
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 26
Guo Y: Local fractional Z transform in fractal space. Adv. Digit. Multimed. 2012, 1(2):96–102.
Yang XJ, Liao MK, Chen JW: A novel approach to processing fractal signals using the Yang-Fourier transforms. Proc. Eng. 2012, 29: 2950–2954.
Yang XJ: Theory and applications of local fractional Fourier analysis. Adv. Mech. Eng. Appl. 2012, 1(4):70–85.
He JH: Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal. 2012., 2012: Article ID 916793
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.
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.2008.08.050
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors would like to thank to the referees for their very useful comments and remarks.
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Yang, XJ., Baleanu, D. & Tenreiro Machado, J.A. Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis. Bound Value Probl 2013, 131 (2013). https://doi.org/10.1186/1687-2770-2013-131
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DOI: https://doi.org/10.1186/1687-2770-2013-131