APPLICATION OF Q-CALCULUS IN QUANTUM GEOMETRY

ABSTRACT

Every geometry is associated with some kind of space. Non-commutative geometry or quantum geometry deals with quantum spaces, including the classical concept of space as a very special case. We consider in particular the case that deals with calculus without limits (quantum calculus); employing the basic governing rules to obtain the q-derivative of some standard functions such as the trigonometric, exponential, logarithmic and hyperbolic functions. We discover that the q-derivative of these functions collapse naturally to the Newton-Leibnitz derivatives. We also considered q-integral which is the inverse of the q-derivative. The Reduced q-Differential Transform Method is presented for solving Partial q-Differential Equations, and the result obtained shows that this iteration procedure is less complicated and efficient when compared with the classical means of obtaining the analytical solution.

TABLE OF CONTENTS

Title page                                                                                                                i  

Declaration                                                                                                              ii

Certification                                                                                                            iii

Dedication                                                                                                              iv

Acknowledgements                                                                                                v

Table of contents                                                                                                    vi

Abstract                                                                                                                  ix

CHAPTER 1

INTRODUCTION                                                                                                    1

1.1  Background of the Study                                                                                   1

1.2  Motivation                                                                                                          3

1.3  Statement of the Problem                                                                                   3

1.4  Aim and Objectives of the Study                                                                       4

1.5  Significance of Study                                                                                         4

1.6  Scope of the Study                                                                                             5

1.7  Limitations                                                                                                          5

CHAPTER 2

REVIEW OF RELATED LITERATURE                                                              6

2.1 Introduction                                                                                                        6

CHAPTER 3

METHODOLOGY                                                                                                    11

3.1 Introduction                                                                                                          11

3.2 Definition (Metric Space)                                                                                     11

3.3 Definition (Cauchy Sequence)                                                                              12

3.3.1 Remark                                                                                                               13

3.4 Banach Space of Continuous Functions                                                               13

3.4.1 Definition                                                                                                           13

3.4.2 Definition                                                                                                           14

3.5 Hilbert Spaces                                                                                                       14

3.6 Mathematical Foundations of Quantum Mechanics                                             15

3.7 Axioms of Quantum Mechanics                                                                           15

3.7.1 Remark                                                                                                               16

3.8 Quantum Geometry                                                                                              17

3.9 Reformulating Basic Geometrical Concepts                                                         17

3.10 The Gelfand-Naimark Theorem                                                                          17

3.11 The Quantum Plane                                                                                            18

3.12 q-Calculus                                                                                                           19

3.13 The q-Differential Operator                                                                                19

3.13.1 Remark                                                                                                             20

3.13.2 Proposition                                                                                                       20

3.14 Taylor Series                                                                                                       22

CHAPTER 4

RESULTS AND DISCUSSIONS                                                                             23

4.1 Basic Notions of q-Calculus                                                                                 23

4.2 q-Numbers and q-Factorials                                                                                  23

4.2.1 Proposition                                                                                                         23

4.2.2 Proposition                                                                                                         26

4.3 q-Binomial Coefficients                                                                                        27

4.3.1 Remark                                                                                                               27

4.3.2 Proposition                                                                                                         27

4.4 q-Derivative of Some Standard Functions                                                           28

4.4.1 q-Derivative of the trigonometric functions                                                      28

4.4.2 q-Derivative of the exponential functions                                                         30

4.4.3 q-Derivative of the logarithmic functions                                                          31

4.4.4 q-Derivative of the hyperbolic functions                                                           31

4.5 q-Integral Operators                                                                                              33

4.5.1 Definition                                                                                                           33

4.5.2 Remark                                                                                                               33

4.5.3 Derivation of The q-Antiderivative                                                                   33

4.5.4 Definition                                                                                                           34

4.5.5 Properties of The q-Integral                                                                               35

4.5.6 Theorem                                                                                                             36

4.5.7 Theorem                                                                                                             36

4.6 The Definite q-Integral                                                                                         38

4.6.1 Definition                                                                                                          38

4.6.2 Remark                                                                                                               39

4.6.3 Remark                                                                                                               40

4.7 Reduced q-Differential Transform Method                                                         40

4.7.1 Definition                                                                                                           40

4.7.2 Definition                                                                                                           41

4.7.3 Definition                                                                                                           41

4.7.4 Theorem                                                                                                             42

4.7.5 Theorem                                                                                                             42

4.7.6 Theorem                                                                                                             42

4.7.7 Theorem                                                                                                             43

4.8 Solution of Partial q-Differential Equation                                                          43

 CHAPTER 5

SUMMARY AND CONCLUSIONS                                                                                   46

5.1 Summary                                                                                                               46

5.2 Conclusion                                                                                                            47

REFERENCES                                                                                                          48

              CHAPTER 1

         INTRODUCTION

1.1 BACKGROUND OF THE STUDY

Quantum mechanics is the science of the microscopic. It is the body of scientific principles that explains the behaviour of matter and its interactions with energy on the scale of atoms and subatomic particles.                                                 .                                  

Classical physics tries to explain matter and energy on a scale familiar to human experience, which includes the behaviour of astronomical bodies. It remains a very important tool in measurement for many aspects of modern science and technology. However, in the 19th century, scientists discovered a lot of phenomena in both the macro and the micro worlds that classical physics could not explain.  In the cause of coming to terms with these limitations, two major revolutions in physics emerged, which created a diversion in the original scientific paradigm thus: the theory of relativity and the development of quantum mechanics.  Physicists became enlightened on the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. These concepts are described in the order in which they were first discovered.

The word quantum is a Latin word that stands for “amount” and in modern idea, means the minimum amount of any physical entity involved in an interaction, for example a packet or quantum of energy. It turns out that certain characteristics of matter can take only discrete values.

Light in some respects behaves like particles and as well behaves like waves in some other respects. Particles such as electrons and atoms exhibits wavelike properties too while some light sources which includes neon lights for instance, give off only certain discrete frequencies of light. Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its energies, colours, and spectral intensities.

Some aspects of quantum mechanics can seem contrary to intuition because they describe the behaviour of matter in a different way from that which is seen in a general context. For example, the uncertainty principle of quantum mechanics means that the more one tries to measure accurately the position of a particle moving in space, one fails in getting the accurate measurement for its momentum simultaneously. 

The word quantum comes from the Latin quantus, meaning how much. Quanta, short for quanta of electricity (electrons) was used in a 1902 article on the photoelectric effect by Philipp Lenard, who credited Hermann von Helmholtz for using the word in the area of electricity. However, the word quantum in general was well known before 1900.Helmholtz used quantum making reference to heat in his article on Mayer's work, which shows that indeed, the word “quantum” can be found in the formulation of the first law of thermodynamics according to Mayer July 24, 1841. Max Planck used quanta to mean quanta of matter and electricity, gas, and heat. Albert Einstein in 1905, in response to Planck's work and the experimental work of Lenard (who explained his results by using the term quanta of electricity), suggested that radiation existed in spatially localized packets which he called quanta of light

The concept of quantization of radiation was discovered in 1900 by Max Planck, who had been trying to understand the emission of radiation from heated objects, known as black-body radiation. This accounted for the fact that certain body changes colour once heated up. He assumed that energy can only be absorbed or released in tiny, discrete packets called bundles or energy elements. Planck reported his revolutionary findings to the German Physical Society on December 14, 1900, and introduced the idea of quantization for the first time as a part of his research on black-body radiation. As a result of his research and experiments, he deduced the numerical value of h, known as the Planck’s constant. After his theory was validated, Planck was awarded the Nobel Prize in Physics in 1918 for his discovery.

1.2 MOTIVATION

We are motivated to go into this study by the need to understand the basis of non-commutative or quantum geometry and its applications. Specifically, the notion of calculus in this non-commutative setting could open up a whole new vista of research into quantum calculus, and our present research effort can be considered a stepping stone towards this direction.

1.3 STATEMENT OF THE PROBLEM

 In this research work our problem is two-fold.

(i)                 We first introduce the  q-differential operator Dq , given by;

then use this to compute the q-derivatives of some standard functions calculus such as  etc.

(ii)        Secondly, we introduce the q-anti-derivative operator , also known as the Jackson integral, and study its properties. It is given by;

1.4 AIM AND OBJECTIVES OF THE STUDY

The aim of this work is to show that one can perform algebraic geometry in a space devoid of points and where the coordinates do not commute. The objectives of this work are

·         To study the basic rules governing q-calculus as compared with the classical Newton-Leibnitz calculus

·         To translate geometry into a commutative algebra format.

·         To introduce the q-differential operator, and use it to deduce the q-derivatives of some classical functions and compare with the results of Newton Leibnitz.

·         To derive the q-integral counterpart of the q-differential operator, referred to as the Jackson Integral.

·         To obtain a solution to some Partial q-differential equations.

1.5 SIGNIFICANCE OF STUDY

This research work is significant because it has important applications in different mathematical areas such as number theory, combinatorics and orthogonal polynomials; but most importantly in the domain of theoretical physics such as quantum mechanics.

1.6 SCOPE OF THE STUDY

The scope of this study covers the preliminary aspect of quantum mechanics which serves as a foundation for this research work. We begin with the fundamental laws of quantum mechanics,  with natural domain in a complex infinite dimensional Hilbert space. The concept of state and observables are emphasized leading us to the so-called Heisenberg’s uncertainty principle. The evolution of a quantum system is described by the famous Shrodinger’s equation. More importantly we considered the notion of q-calculus involving q-derivative and q-antiderivative of standard classical functions such as monomials, exponential functions, logarithmic function, trigonometric and hyperbolic functions. We consider an application of this study to the solution of heat equation.

1.7 LIMITATIONS

This research work was limited to the computation of q-derivatives and q-antiderivatives (Jackson integral) of classical functions and their properties. The application of this q-calculus was limited the solution of Partial q-Differential Equation.

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