The book “Fundamentals of Analysis in Physics‘ is aimed at undergraduate students who will be learning the basics of physics. Many beginners find it difficult to learn each area of ​​physics (classical mechanics, electromagnetism, quantum mechanics, relativistic quantum mechanics, and statistical mechanics) in detail individually and in detail. It would be preferable to learn all fields as quickly as possible and have a simple idea about the relationships between different fields. After you learn the location of each box in physics, it becomes easier to learn detailed parts of each box. In this book, the important points of all branches of physics are summarized with short and simple expressions as follows.

1. The differential and integrals are important in understanding physics but may not be familiar to all readers. The essential parts are summarized.
2. Classical mechanics is based on Newton’s three laws, the descriptions of which are simple. But it is not easy to solve the actual equations of motion. The basics of transforming coordinates to simplify the equations are introduced.
3. Electromagnetism is summed up by Maxwell’s equations. They are modifications of descriptions of already known laws. It explains why it started a revolution in physics: the identity of light as an electromagnetic wave was clarified. The basics of the theory of relativity are also introduced (e.g. derivation of the rest energyE=mC2).
4. The fundamentals of quantum mechanics are introduced from the analogy of the 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. Photon energy and momentum are given by frequency and wavenumber: this relationship should be common to all particles. The Schrödinger equation was derived from the relationship between energy and momentum given by classical mechanics. In this book, solutions of the Schrödinger equation are described with simplified descriptions without using special functions.
5. The wave function including the electron spin should be given by a vector and is not derived from the Schrödinger equation using a scalar formula. On the other hand, a formula using matrices (Dirac’s equation) was required to express the relationship between energy and momentum given by the theory of relativity. The characteristic of the electron spin in a magnetic field was also derived from the Dirac equation.

Chapter 3 describes the principle of the HF ion trap without using the Mathieu equation. In Chapter 4, the basics of electrically induced transparency (EIT) and adiabatic fast passage are explained in simple terms.

dr Masatoshi Kajita was born and raised in Nagoya, Japan. He graduated from the Department of Applied Physics, University of Tokyo in 1981 and received his PhD from the Department of Physics, University of Tokyo in 1986. After working at the Institute of Molecular Sciences, he joined the Communications Research Laboratory (CRL) in 1989. In 2004, the CRL was renamed the National Institute of Information and Communications Technology (NICT). At NICT he focused on the precision measurement of atomic transition frequencies. Since 2008 he has been interested in the precision measurement of vibrational transition frequencies of molecules. In 2009 he was a visiting professor at Provence University, Marseille, France. As of 2021, he has published 91 research articles and 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 calculations, Schrödinger equation, classical physics, energy structure of atoms and molecules, two-body system, electron spin, Lagrange equation, Dirac equation 