#### Fundamentals of Mathematical Treatments

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Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010002

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##### Abstract

#### Analysis in the Classical Mechanics

Page: 21-32 (12)

Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010003

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##### Abstract

#### Fundamental Meaning and Typical Solutions of Maxwell’s Equations

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Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010004

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##### Abstract

#### Fundamentals of Analysis in Quantum Mechanics

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Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010005

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##### Abstract

Light has the dual characteristics of particles (photons) and electromagnetic waves. The photon has an energy of 𝐸 = ℎ𝜈 (𝜈: frequency, h: Planck constant) and the momentum of 𝑝⃗ = ℎ𝑘⃗⃗ (𝑘⃗⃗: wavenumber and 1|𝑘⃗⃗| is the wavelength). The photon density is proportional to the square of the amplitude of the electromagnetic waves. The fundamental aspect of quantum mechanics is that these characteristics apply to all matters. The properties of matters are described by wave functions. The probability of the existence of the matter is proportional to the square of the associated wave function. When a matter is localized in a limited region, it can only assume discrete values of energy because the wavelength of the matter wave must be an integral division of the region. The phase of the wavefunction has uncertainty on order 1/2 radians; therefore, position and momentum (time and energy) cannot be simultaneously determined. As the size of the localization area of the wavefunction becomes smaller, the minimum kinetic energy becomes larger because of the smaller wavelength (larger momentum uncertainty). The Schroedinger equation was derived based on the idea that the relationship between the frequency and the wavenumber corresponds to that between energy and momentum given by classical mechanics, which makes it possible to obtain the wave functions of matters in the energy eigenstates. Several examples of solutions to the Schroedinger equation are introduced. The mixture between different energy eigenstates and the shift in the energy eigenvalues are induced by electromagnetic fields. The temporal change of the wave function (transition between different energy states) is also obtained using the Schroedinger equation.

#### Relativistic Quantum Mechanics and Spin

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Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010006

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##### Abstract

*1/2*spin states and the positive and negative rest energies. The existence

#### Fundamentals of Statistical Mechanics Using the Boltzmann Distribution

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Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010007

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##### Abstract

#### Analysis of the Measurement Uncertainties

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Author: Masatoshi Kajita

DOI: 10.2174/9789815049107122010008

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##### Abstract

Physical laws, which have finite uncertainties, are established to facilitate measurements. New physical phenomena have been discovered when the measurement uncertainties were reduced. The object of this chapter is to review the estimation of the measurement uncertainties. This parameter consists of statistical uncertainty and systematic uncertainty. Statistical uncertainty is given by the random distribution of measurement results in a region with a broadening of 𝜎 in the vicinity of a real value. The statistical uncertainty of the average of the measurement results is expected to be 𝜎√𝑁, where N is the number of measurement samples. Systematic uncertainty exists because measurements are influenced by the conditions under which they are obtained. The real value is defined for a certain circumstance, and the measurements obtained under different circumstances are shifted from this value. The real value is obtained by correcting the shift, which is estimated according to the measurement circumstances. Statistical uncertainty is obtained from the uncertainty in the estimation of the shifts.

## Introduction

Many beginners find physics to be a challenging subject to learn, and the difficulty extends to each branch of physics. It would be preferable for beginners to learn about different branches of physics as quickly as possible with a simplified understanding of the relevant mathematical relationships. After learning the position of each field in physics, it becomes easier to learn details of each field. In this book, special functions are not used to explain the solutions of equations. Fundamentals of Analysis In Physics summarizes the analytical methods in different fields of physics The book covers several known fields of physics and is a useful text for beginners in physics, college and university students, and working professionals who may not have a background in mathematics or physics. Key features: - Summarizes information about different fields in physics in 150 pages - Covers 7 different fields of physics (classical mechanics, electromagnetism, quantum mechanics, relativistic quantum mechanics, statistical mechanics and more) in 7 separate chapters -Contains simple explanations without the use of special functions.