The book “** Fundamentals of analysis in physics**” is targeting undergraduate students, who are going to learn the fundamentals of physics. Many beginners feel that it is difficult to learn each field of physics (classical mechanics, electromagnetism, quantum mechanics, relativistic quantum mechanics, and 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 important points of all fields of physics are summarized with short and simple expressions as follows.

- The differential and integrals are important to understand physics, but might not be familiar for all readers. The fundamental parts are summarized.
- The classical mechanics is based on the Newton’s three laws, whose descriptions are simple. But it is not simple to solve the actual motion equations. The fundamentals of transform of coordinates to simplify the equations are introduced.
- The electro-magnetism are summarized by Maxwell’s equations. They are change of descriptions of laws which were already known. It is explained why it caused a revolution in physics: the identity of light as an electromagnetic wave was clarified. The fundamental of theory of relativity is also introduced (for example, derivation of rest energy
*E=m**c**2*). - The fundamental of quantum mechanics is introduced from the analogy of duality of light (electromagnetic wave, photon). The photon density is proportional to the square of the amplitude of the electromagnetic wave, then the density of the particle should be proportional to the square of the amplitude of the matter wave. The photon energy and the momentum are given by the frequency and wavenumber: this relation should be common to all particles. From the relation between energy and momentum given by classical mechanics, Schroedinger equation was derived. In this book, solutions of Schrodinger equation are described with simplified descriptions without using special functions.
- The wave function including the electron spin should be given by a vector, and it is not derived from the Schrodinger equation with a scalar formula. On the other hand, a formula with matrices (Dirac equation) was required to give the relation between energy and momentum given by the theory of relativity. The characteristic of electron spin in a magnetic field was also derived from the Dirac equation.

In chapter 3, the principle of rf-ion trap is described without using the Mathieu-equation. In chapter 4, the fundamentals of electric induced transparency (EIT) and adiabatic rapid passage are explained with simple expressions.

**About the Editor:**

Dr. Masatoshi Kajita was born and raised in Nagoya, Japan. He graduated from the Department of Applied Physics, the University of Tokyo, in 1981 and obtained his Ph. D. from Department of Physics, the University of Tokyo, in 1986. After working at the Institute for Molecular Science, he joined Communications Research Laboratory (CRL) in 1989. In 2004, the CRL was renamed the National Institute of Information and Communications Technology (NICT). At NICT, he has been focused to the precision measurement of atomic transition frequencies. Since 2008, he has been interested with the precision measurement of vibrational transition frequencies of molecules. In 2009, he was a guest professor at the Provence University, Marseille, France. Until 2021, he has published 91 research articles and following three books: “Measuring Time; Frequency measurements and related developments in physics (2018, IOP Expanding physics)”, “Measurement, Uncertainty and Lasers (2019, IOP Expanding Physics)”, and “Cold Atoms and Molecules (2021, IOP Expanding Physics)”.

**Keywords:**

Physical mathematics, Quantum mechanics, Numerical calculation, Schrodinger equation, Classical physics, Energy structure of atoms and molecules, Two body system, Electron spin , Lagrange equation, Dirac equation

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