Fundamentals of Analysis in Physics

159 Pages·2022·11.829 MB·English

by  Masatoshi Kajita

Fundamentals of Analysis
in Physics

Authored by
Masatosh i Kajita
National Institute of Information and Communications Technology,
Koganei, Tokyo, Japan
Fundamentals of Analysis in Physics
Author: Masatoshi Kajita
ISBN (Online): 978-981-5049-10-7
ISBN (Print): 978-981-5049-11-4
ISBN (Paperback): 978-981-5049-12-1
©2022, Bentham Books imprint.
Published by Bentham Science Publishers Pte. Ltd. Singapore. All Rights Reserved.
BENTHAM SCIENCE PUBLISHERS LTD.
End User License Agreement (for non-institutional, personal use)
This is an agreement between you and Bentham Science Publishers Ltd. Please read this License Agreement
carefully before using the ebook/echapter/ejournal (“Work”). Your use of the Work constitutes your
agreement to the terms and conditions set forth in this License Agreement. If you do not agree to these terms
and conditions then you should not use the Work.
Bentham Science Publishers agrees to grant you a non-exclusive, non-transferable limited license to use the
Work subject to and in accordance with the following terms and conditions. This License Agreement is for
non-library, personal use only. For a library / institutional / multi user license in respect of the Work, please
contact: [email protected].
Usage Rules:
1. All rights reserved: The Work is the subject of copyright and Bentham Science Publishers either owns the
Work (and the copyright in it) or is licensed to distribute the Work. You shall not copy, reproduce, modify,
remove, delete, augment, add to, publish, transmit, sell, resell, create derivative works from, or in any way
exploit the Work or make the Work available for others to do any of the same, in any form or by any
means, in whole or in part, in each case without the prior written permission of Bentham Science
Publishers, unless stated otherwise in this License Agreement.
2. You may download a copy of the Work on one occasion to one personal computer (including tablet,
laptop, desktop, or other such devices). You may make one back-up copy of the Work to avoid losing it.
3. The unauthorised use or distribution of copyrighted or other proprietary content is illegal and could subject
you to liability for substantial money damages. You will be liable for any damage resulting from your
misuse of the Work or any violation of this License Agreement, including any infringement by you of
copyrights or proprietary rights.
Disclaimer:
Bentham Science Publishers does not guarantee that the information in the Work is error-free, or warrant that
it will meet your requirements or that access to the Work will be uninterrupted or error-free. The Work is
provided "as is" without warranty of any kind, either express or implied or statutory, including, without
limitation, implied warranties of merchantability and fitness for a particular purpose. The entire risk as to the
results and performance of the Work is assumed by you. No responsibility is assumed by Bentham Science
Publishers, its staff, editors and/or authors for any injury and/or damage to persons or property as a matter of
products liability, negligence or otherwise, or from any use or operation of any methods, products instruction,
advertisements or ideas contained in the Work.
Limitation of Liability:
In no event will Bentham Science Publishers, its staff, editors and/or authors, be liable for any damages,
including, without limitation, special, incidental and/or consequential damages and/or damages for lost data
and/or profits arising out of (whether directly or indirectly) the use or inability to use the Work. The entire
liability of Bentham Science Publishers shall be limited to the amount actually paid by you for the Work.
General:
1. Any dispute or claim arising out of or in connection with this License Agreement or the Work (including
non-contractual disputes or claims) will be governed by and construed in accordance with the laws of
Singapore. Each party agrees that the courts of the state of Singapore shall have exclusive jurisdiction to
settle any dispute or claim arising out of or in connection with this License Agreement or the Work
(including non-contractual disputes or claims).
2. Your rights under this License Agreement will automatically terminate without notice and without the
need for a court order if at any point you breach any terms of this License Agreement. In no event will any
delay or failure by Bentham Science Publishers in enforcing your compliance with this License Agreement
constitute a waiver of any of its rights.
3. You acknowledge that you have read this License Agreement, and agree to be bound by its terms and
conditions. To the extent that any other terms and conditions presented on any website of Bentham Science
Publishers conflict with, or are inconsistent with, the terms and conditions set out in this License
Agreement, you acknowledge that the terms and conditions set out in this License Agreement shall prevail.
Bentham Science Publishers Pte. Ltd.
80 Robinson Road #02-00
Singapore 068898
Singapore
Email: [email protected]
CONTENTS
PREFACE ................................................................................................................................................... i
CHAPTER 1 FUNDAMENTALS OF MATHEMATICAL TREATMENTS ....................................... 1
INTRODUCTION ............................................................................................................................. 1
ITERATIVE SOLUTION OF EQUATIONS.................................................................................. 2
DIFFERENTIAL AND INTEGRAL EQUATIONS ...................................................................... 3
FIRST-ORDER DIFFERENTIAL EQUATIONS .......................................................................... 7
SECOND-ORDER DIFFERENTIAL EQUATIONS ..................................................................... 8
NUMERICAL CALCULATION OF DIFFERENTIAL EQUATIONS ....................................... 10
CHAOS ............................................................................................................................................... 15
EIGENVALUES AND EIGENVECTORS OF MATRICES ........................................................ 17
EXERCISE ........................................................................................................................................ 19
REFERENCES .................................................................................................................................. 20
CHAPTER 2 ANALYSIS IN THE CLASSICAL MECHANICS .......................................................... 21
FUNDAMENTAL OF CLASSICAL MECHANICS ...................................................................... 21
TWO-BODY MOTION AND TORQUE ......................................................................................... 23
LAGRANGE EQUATION ............................................................................................................... 26
APPLICATION OF LAGRANGE EQUATION TO ELECTROMAGNETIC FIELDS ............ 29
EXERCISE ........................................................................................................................................ 31
REFERENCES .................................................................................................................................. 32
CHAPTER 3 FUNDAMENTAL MEANING AND TYPICAL SOLUTIONS OF MAXWELL’S
EQUATIONS ............................................................................................................................................... 33
WHAT ARE MAXWELL’S EQUATIONS? .................................................................................. 33
WHY WERE MAXWELL’S EQUATIONS REVOLUTIONARY? ............................................. 37
Can we Trap Charged Particles? .............................................................................................. 37
Distribution of the Electric Field .............................................................................................. 40
ENERGY OF ELECTROMAGNETIC FIELDS ........................................................................... 41
ELECTROMAGNETIC WAVE AS THE IDENTITY OF LIGHT ............................................. 42
DESCRIPTION OF MAXWELL’S EQUATIONS USING THE VECTOR POTENTIAL ....... 45
SIMPLE INTERPRETATION OF THE THEORY OF SPECIAL RELATIVITY ................... 48
EXERCISE ........................................................................................................................................ 56
REFERENCES ................................................................................................................................. 57
CHAPTER 4 FUNDAMENTALS OF ANALYSIS IN QUANTUM MECHANICS ............................. 58
ESTABLISHMENT OF QUANTUM MECHANICS ..................................................................... 58
FUNDAMENTALS OF QUANTUM MECHANICS ..................................................................... 61
SOLUTION OF ENERGY EIGENVALUES .................................................................................. 63
Solution for a One-dimensional Potential Well ........................................................................ 64
Harmonic Potential .................................................................................................................. 66
SCHROEDINGER EQUATION IN THREE-DIMENSIONAL POLAR COORDINATES ...... 67
Angular Momentum ................................................................................................................. 68
Wave Function in Free Space Using the Polar Coordinate ....................................................... 71
Energy State of an Electron in a Hydrogen Atom .................................................................... 73
Energy State of Diatomic Molecules ........................................................................................ 76
SLIGHT ENERGY SHIFT BY PERTURBATIO CHANGE OF THE ENERGY STATE
WHEN THERE IS TEMPORAL CHANGE IN THE HAMILTONIANN .................................. 83
Sudden and Slow Change of the Potential ............................................................................... 83
The Transition Induced by the AC Electric Field ...................................................................... 84
Electric Induced Transparency (EIT) ....................................................................................... 88
State Mixture Induced by the AC Electric Field (Light) .......................................................... 90
QUANTUM TREATMENT OF ELASTIC COLLISION BY A SPHERICALLY
EXERCISE ........................................................................................................................................ 95
REFERENCES .................................................................................................................................. 96
CHAPTER 5 RELATIVISTIC QUANTUM MECHANICS AND SPIN ............................................... 97
ELECTRON SPIN............................................................................................................................. 97
KLEIN-GORDON EQUATION ...................................................................................................... 100
DIRAC EQUATION AND DERIVATION OF ELECTRON SPIN ............................................. 100
RELATIVISTIC CORRECTION USING THE DIRAC EQUATION ........................................ 104
MEANING OF NEGATIVE REST ENERGY ............................................................................... 109
UTILITY OF THE EQUATIONS FOR WAVE FUNCTIONS WITH DIFFERENT SPIN ....... 110
EXERCISE ........................................................................................................................................ 110
REFERENCES .................................................................................................................................. 110
CHAPTER 6 FUNDAMENTALS OF STATISTICAL MECHANICS USING THE BOLTZMANN
DISTRIBUTION ......................................................................................................................................... 112
THERMAL ENERGY ...................................................................................................................... 112
BOLTZMANN DISTRIBUTION ..................................................................................................... 113
AVERAGE OF ENERGY ................................................................................................................. 115
CHANGE OF THE TOTAL ENERGY BY CHANGING THE POTENTIAL PARAMETER . 119
THE SPECIFIC HEAT OF GASEOUS AND SOLID ATOMS AND MOLECULES ................ 119
POPULATION DISTRIBUTION WHEN EVENTS ARE NOT INDEPENDENT ..................... 122
THE IDEAL GAS LAW AND THE WORK DONE BY THE GAS PRESSURE ........................ 125
EXERCISE ........................................................................................................................................ 132
REFERENCES ................................................................................................................................. 133
CHAPTER 7 ANALYSIS OF THE MEASUREMENT UNCERTAINTIES…………………………… 134
IMPORTANCE OF THE MEASUREMENT UNCERTAINTY................................................... 134
STATISTICAL AND SYSTEMATIC UNCERTAINTIES ............................................................ 137
Statistical Uncertainty .............................................................................................................. 138
Systematic Uncertainty ............................................................................................................ 140
TOTAL UNCERTAINTY ................................................................................................................ 141
EXERCISE ........................................................................................................................................ 142
REFERENCES ................................................................................................................................. 142
CONCLUSION ........................................................................................................................................... 144

SUBJECT INDEX .................................................................................................................................... 1(cid:23)(cid:24)
i
PREFACE
This book summarizes the analysis in whole fields of physics without using the
special functions, targeting college/bachelor students. Many beginners feel that it
is difficult to learn each field of physics (classical mechanics, electromagnetism,
quantum mechanics, relativistic quantum mechanics, statistic mechanics) in detail
separately. It would be preferable to learn the whole fields as quick as possible
and have a simple imagination about the relation between different fields. After
learning the position of each field in the physics, it becomes easier to learn
detailed parts of each field. In this book, the fundamental of mathematical
treatments are introduced, which are important for the analysis in physics but not
familiar to all the readers. The fundamental of the analysis in each field of physics
are summarized afterwards. The estimation of measurement uncertainty is also
introduced. The important points of whole physics are summarized within 150
pages.

It would be my great pleasure, if this book can help college students to understand
the fundamental of physics. And I believe, it is also useful for researchers to
develop new research fields.

CONSENT FOR PUBLICATION
Not applicable.
CONFLICT OF INTEREST
The author confirms that there is no conflict of interest.
ACKNOWLEDGEMENTS

The research activity of the author is supported by a Grant-in-Aid for Scientific
Research (B) (Grant No. JP 17H02881 and JP20H01920) and a Grant-in-Aid for
Scientific Research (C) (Grant Nos. JP 17K06483 and 16K05500) from the Japan
Society for the Promotion of Science (JSPS). I want to thank Editage
(www.editage.com) for English language editing.

Masatoshi Kajita
National Institute of Information and Communications
Technology, Koganei,
Tokyo, Japan
Fundamentals of Analysis in Physics, 2022, 1-20 1
CHAPTER 1

Fundamentals of Mathematical Treatments
Abstract: Physical phenomena can be understood by solving equations that lead to
physical laws. The first objective of this chapter is to solve certain equations that are
required for physical analysis. First, an iterative solution of the equation is introduced.
Using this approach, the numerical solution of an equation f(x) = 0 can be obtained
also when the function f(x) is too complicated for the solution to be obtained as an
explicit formula.
Many physical equations can be expressed using differential and integral
mathematical representations, which might not be familiar to all college students. The
fundamental concepts of the differential and integral were introduced. Several
fundamental mathematical formulae are reviewed.
The second objective is to solve the differential equations that are required for the
physical analysis. First, some solutions of simple differential equations given by
explicit formulas are introduced, which are important for their physical interpretation.
However, the equations for technical use are generally too complicated. Several
methods for obtaining numerical solutions are introduced, which are useful for
analyzing motion orbits. There is also a phenomenon that cannot be predicted by
solving equations, which is called “chaos.”
Finally, the fundamentals of the eigenvalues of matrices are introduced, which are
important for understanding quantum mechanics. This chapter was prepared for
undergraduate students who are not familiar with differential and integral calculus and
matrices.
Keywords: Iterative solution, Differential, Partial derivative, Integral, Taylor
expansion, Euler method, Middle point method, Runge-Kutta method, Chaos,
Lyapunov exponent, matrix, Determinant, Eigen value, Eigen vector.
1.1. INTRODUCTION

Many physical laws have been established based on equations, and physical
phenomena are predicted by solving them. It is not always a simple task to solve
these equations. For example, the Newtonian law of universal gravitation is given

Masatoshi Kajita
All rights reserved-© 2022 Bentham Science Publishers
2 Fundamentals of Analysis (cid:76)(cid:81) Physics Masatoshi Kajita
by the simple formula. However, it is not easy to analyze the motion of astronomical
bodies when considering the interaction between more than three bodies.

The technical development of any analysis plays an important role in physics. For
some cases, the simplification of equations makes it possible to obtain a solution
that is expressed as simple formulas, which can provide new physical insights.
However, numerical calculations make it possible to obtain the solutions of
equations which cannot be expressed as a rigorous formula. The technical
development of numerical calculations is important to minimize the error of
solutions and the calculation time.

Physicists and engineers have many techniques for obtaining reliable solutions to
equations. Different kinds of approximations have been developed in the field of
theoretical physics, which are not acceptable for mathematicians. This chapter
summarizes the fundamentals of solving the equations analytically and numerically.

Reference [1] seems to be useful to learn the fundamental of mathematics more in
detailed for students, who are interested with physical analysis. Reference [2] seems
to be readable also for high school students [2].

1.2. ITERATIVE SOLUTION OF EQUATIONS

There are many cases for which the solution x that satisfies the following equation
0
must be obtained:

𝑓(𝑥 ) = 0 (1.2.1)
0

Considering the following simple equations, the solutions are given by:

𝑏
𝑓(𝑥) = 𝑎𝑥 +𝑏 → 𝑥 = − (1.2.2)
0
𝑎
𝑓(𝑥) = 𝑎𝑥2 +𝑏𝑥 +𝑐 → 𝑥 = −𝑏±√𝑏2−4𝑎𝑐 (1.2.3)
0
2𝑎

In many other cases, the solutions cannot be described using simple formulas.
Therefore, solutions are often obtained using an iterative method. When 𝑓(𝑥 ) >
𝑎
0 𝑓(𝑥 ) < 0 with 𝑥 < 𝑥 , the solution x is 𝑥 < 𝑥 < 𝑥 . Then we calculate
𝑏 𝑎 𝑏 0 𝑎 0 𝑏
𝑓(𝑥 ) with 𝑥 < 𝑥 < 𝑥 . If 𝑓(𝑥 ) > 0, 𝑥 < 𝑥 < 𝑥 is obtained. By repeating
𝑐 𝑎 𝑐 𝑏 𝑐 𝑐 0 𝑏