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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis

Xiao-Jun Yang123*, Dumitru Baleanu456 and José António Tenreiro Machado7

Author Affiliations

1 Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu, 221008, China

2 Institute of Software Science, Zhengzhou Normal University, Zhengzhou, 450044, China

3 Institute of Applied Mathematics, Qujing Normal University, Qujing, 655011, China

4 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, 06530, Turkey

5 Institute of Space Sciences, Magurele, Bucharest, Romania

6 Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

7 Department of Electrical Engineering, Institute of Engineering of Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida, 431, Porto, 4200-072, Portugal

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Boundary Value Problems 2013, 2013:131  doi:10.1186/1687-2770-2013-131

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/131


Received:29 January 2013
Accepted:2 May 2013
Published:20 May 2013

© 2013 Yang et al.; licensee Springer

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.

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 time-space

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

Definition 1[18-20,27-30]

If there is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M1">View MathML</a>

(2.1)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M2">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M4">View MathML</a>. Now <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a> is called a local fractional continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M6">View MathML</a>, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M7">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a> is called local fractional continuous on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M9">View MathML</a>, denoted by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M10">View MathML</a>

(2.2)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a> is said to be local fractional continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M12">View MathML</a> from the right if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M13">View MathML</a> is defined, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M14">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a> is said to be local fractional continuous at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M12">View MathML</a> from the left if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M13">View MathML</a> is defined, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M18">View MathML</a>

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M21">View MathML</a>, then we have the following relation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M22">View MathML</a>

For other results of theory of local fractional continuity of functions, see [18-20,27-30].

2.2 Local fractional derivative and integration

Definition 2[18-20,27-30]

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M23">View MathML</a>, a local fractional derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a> of order α at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M6">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M26">View MathML</a>

(2.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M27">View MathML</a> with a gamma function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M28">View MathML</a>.

Definition 3[18-20,27-30]

Setting<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M23">View MathML</a>, a local fractional integral of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M30">View MathML</a> of order α in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M31">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M32">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M38">View MathML</a>, is a partition of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M31">View MathML</a>.

Their fractal geometrical explanation of local fractional derivative and integration can be seen in [22,26,50-52].

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M40">View MathML</a>, then we have [18,19]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M41">View MathML</a>

(2.5)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M42">View MathML</a>.

Lemma 1[18,19]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M43">View MathML</a>

(2.6)

Proof See [18,19]. □

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

Definition 4[18,19,49-52]

The local fractional trigonometric Fourier series of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M44">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M45">View MathML</a>

(3.1)

Then the local fractional Fourier coefficients can be computed by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M46">View MathML</a>

(3.2)

The Mittag-Leffler functions expression of the local fractional Fourier series is described by [18,19,49-52]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M47">View MathML</a>

(3.3)

where the local fractional Fourier coefficients are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M48">View MathML</a>

(3.4)

The above is generalized to calculate the local fractional Fourier series.

3.2 The Fourier transform in fractal space

Definition 5[18,19,49-53]

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M49">View MathML</a>, the Fourier transform in fractal space, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M50">View MathML</a>, is written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M51">View MathML</a>

(3.5)

where the latter converges.

Definition 6[18,19,49-53]

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M52">View MathML</a>, its inversion formula is written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M53">View MathML</a>

(3.6)

3.3 The generalized Fourier transform in fractal space

Definition 7[18,19]

The generalized Fourier transform in fractal space is written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M54">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M55">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M56">View MathML</a>.

Definition 8[18,19]

The inverse formula of the generalized Fourier transform in fractal space is written in the form [18,19]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M57">View MathML</a>

(3.8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M55">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M59">View MathML</a>.

3.4 Some useful results

The following formula is valid [18,19].

Theorem 1[18,19]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M60">View MathML</a>

(3.9)

Proof See [18,19]. □

Theorem 2[18,19]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M61">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M62">View MathML</a>

(3.10)

Proof See [18,19]. □

Theorem 3[18,19]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M61">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M64">View MathML</a>

(3.11)

Proof See [18,19]. □

4 Heisenberg uncertainty principles in local fractional Fourier analysis

Theorem 4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M66">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M67">View MathML</a>

(4.1)

with equality only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a>is almost everywhere equal to a constant multiple of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M69">View MathML</a>

(4.2)

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M70">View MathML</a>and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M71">View MathML</a>.

Proof Considering the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M72">View MathML</a>

(4.3)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M73">View MathML</a>

(4.4)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M74">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M75">View MathML</a> with a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M71">View MathML</a>.

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M77">View MathML</a>

(4.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M78">View MathML</a>

(4.6)

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M79">View MathML</a>

(4.7)

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M81">View MathML</a>.

Hence, there is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M82">View MathML</a>

(4.8)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M83">View MathML</a>

(4.9)

Therefore, we deduce to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M84">View MathML</a>

(4.10)

Hence, this result is obtained. □

As a direct result, we have two equivalent forms as follows.

Theorem 5Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M65">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M86">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M87">View MathML</a>

(4.11)

with equality only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M5">View MathML</a>is almost everywhere equal to a constant multiple of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M89">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M90">View MathML</a>and a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M71">View MathML</a>.

Proof Applying Theorem 4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M92">View MathML</a>

(4.12)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M93">View MathML</a>

(4.13)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M94">View MathML</a>

(5.1)

where the Planck-Einstein and De Broglie relations are in fractal space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M95">View MathML</a>

(5.2)

We can realize the local fractional partial derivative with respect to fractal space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M96">View MathML</a>

(5.3)

and fractal time

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M97">View MathML</a>

(5.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M98">View MathML</a>[26] with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M99">View MathML</a>[26].

From (5.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M100">View MathML</a>

(5.5)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M101">View MathML</a>

(5.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M102">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M103">View MathML</a>.

From (5.4) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M104">View MathML</a>

(5.7)

We have the energy equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M105">View MathML</a>

(5.8)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M106">View MathML</a>

(5.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M107">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M108">View MathML</a> is the local fractional Hamiltonian in fractal mechanics.

Hence, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M109">View MathML</a>

(5.10)

Therefore, we can deduce that the local fractional energy operator is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M110">View MathML</a>

(5.11)

and that the local fractional momentum operator is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M111">View MathML</a>

(5.12)

Therefore, we get the local fractional Schrödinger equation in the form of local fractional energy and momentum operators

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M112">View MathML</a>

(5.13)

where the local fractional Hamiltonian is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M113">View MathML</a>

(5.14)

We also deduce that the general time-independent local fractional Schrödinger equation is written in the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M114">View MathML</a>

(5.15)

which is related to the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M115">View MathML</a>

(5.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M116">View MathML</a> is non-differential action, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M108">View MathML</a> is the local fractional Hamiltonian function, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M118">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M119">View MathML</a>) 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M120">View MathML</a>

(5.17)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M121">View MathML</a>

(5.18)

If we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M122">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M123">View MathML</a>, we have fractal plane waves:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M124">View MathML</a>

(5.19)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M125">View MathML</a>

(5.20)

In view of (5.20), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M126">View MathML</a>

(5.21)

The set of N measurements is characterized by an expectation value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M127">View MathML</a> and a root mean square dispersion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M128">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M129">View MathML</a>

(5.22)

Similarly, the square of the dispersion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M130">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M131">View MathML</a>

(5.23)

If the physical interpretation of a particle in fractal space is that the probability

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M132">View MathML</a>

(5.24)

the integral of this quantity over all fractal space is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M133">View MathML</a>

(5.25)

For (5.18) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M134">View MathML</a>

(5.26)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M135">View MathML</a>

(5.27)

5.2.3 Probabilistic interpretation of fractal complex wave function of one variable

In (5.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M136">View MathML</a>

(5.28)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M137">View MathML</a>

(5.29)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M138">View MathML</a>

(5.30)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M139">View MathML</a>

(5.31)

We have an expectation value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M140">View MathML</a> and a root mean square dispersion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M141">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M142">View MathML</a>

(5.32)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M143">View MathML</a>

(5.33)

For a given fractal mechanical operatorA, we have an expectation value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M144">View MathML</a> and a root mean square dispersion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M145">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M146">View MathML</a>

(5.34)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M147">View MathML</a>

(5.35)

5.3 The Heisenberg uncertainty principle in fractal quantum mechanics

Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M148">View MathML</a>

(5.36)

we have a fractal positional operator expectation value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M149">View MathML</a>

(5.37)

and a root mean square dispersion of positional operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M150">View MathML</a>

(5.38)

Similar to the fractal positional operator, we have a fractal momentum operator expectation value

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M151">View MathML</a>

(5.39)

and a root mean square dispersion of positional operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M152">View MathML</a>

(5.40)

Considering

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M153">View MathML</a>

(5.41)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M154">View MathML</a>

(5.42)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M155">View MathML</a>

(5.43)

by using Theorem 5, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M156">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M157">View MathML</a>

(5.44)

Hence, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M158">View MathML</a>

(5.45)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M159">View MathML</a>

(5.46)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M160">View MathML</a>

(5.47)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M161">View MathML</a>

(5.48)

Suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M162">View MathML</a>

(5.49)

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M163">View MathML</a>

(5.50)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M164">View MathML</a>

(5.51)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M102">View MathML</a>[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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M166">View MathML</a>

(5.52)

and the local fractional momentum operator

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M167">View MathML</a>

(5.53)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M98">View MathML</a>[26].

Thus, we get the Planck-Einstein and de Broglie relations are in fractal space as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M169">View MathML</a>

(5.54)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M170">View MathML</a>. We reported the structure the local fractional Schrödinger equation derived from Planck-Einstein and de Broglie relations in fractal time space.

Appendix

We have [13,20]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M171">View MathML</a>

(A.1)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M172">View MathML</a>

(A.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M173">View MathML</a> is a fractal integral staircase function. We have the relation [18-20]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M174">View MathML</a>

(A.3)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M175">View MathML</a>

(A.4)

Inversely we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M176">View MathML</a>

(A.5)

Hence, both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M178">View MathML</a> are seen in [20,21].

In view of Eq. (A.4), we easily obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M179">View MathML</a>

(A.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M180">View MathML</a>

(A.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/131/mathml/M184">View MathML</a> is a fractal unit of imaginary number [18-20,53].

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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