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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