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.
Keywords:
Heisenberg uncertainty principle; local fractional Fourier operator; Schrödinger equation; fractal timespace1 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 timespace [310]. The theory of local fractional calculus [1120], started to be considered as one of the useful tools to handle the fractal and continuously nondifferentiable functions. This formalism was applied in describing physical phenomena such as continuum mechanics [21], elasticity [2022], quantum mechanics [23,24], heatdiffusion and wave phenomena [2530], and other branches of applied mathematics [3133] and nonlinear dynamics [34,35].
The fractional Heisenberg uncertainty principle and the fractional Schrödinger equation based on fractional Fourier analysis were proposed [3648]. Local fractional Fourier analysis [49], which is a generalization of the Fourier analysis in fractal space, has played an important role in handling nondifferentiable functions. The theory of local fractional Fourier analysis is structured in a generalized Hilbert space (fractal space), and some results were obtained [26,4953]. 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 [1820,2730].
2.2 Local fractional derivative and integration
Setting , a local fractional derivative of of order α at is defined by
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 [22,26,5052].
3 Theory of local fractional Fourier analysis
In this section, we investigate local fractional Fourier analysis [4953], 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 MittagLeffler functions expression of the local fractional Fourier series is described by [18,19,4952]
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
The inverse formula of the generalized Fourier transform in fractal space is written in the form [18,19]
3.4 Some useful results
The following formula is valid [18,19].
4 Heisenberg uncertainty principles in local fractional Fourier analysis
Theorem 4Suppose that, , then we have
with equality only ifis almost everywhere equal to a constant multiple of
Proof Considering the equality
we have
When , then we have with a constant .
Since
and
we have
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 5Suppose thatand, then we have
with equality only ifis almost everywhere equal to a constant multiple of, withand 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 nondifferential phase of a fractal plane wave as a complex phase factor using the relations
where the PlanckEinstein 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
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 timeindependent local fractional Schrödinger equation is written in the form
which is related to the following equation:
where is nondifferential 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 [3840,54,55].
Below we define the local fractional energy operator
and the local fractional momentum operator
where [26].
Thus, we get the PlanckEinstein 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 nondifferentiable 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 timefrequency analysis in fractal space. From Eq. (A.7) we conclude that there is a semigroup property for the MittagLeffler 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 PlanckEinstein and de Broglie relations in fractal time space.
Appendix
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Authors contributed equally in writing this article. Authors read and approved the final manuscript.
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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