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.
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.
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