On the Novel Aspects of 'Introduction to Modern Quantum Mechanics: Focusing on Quantum Information and Quantum Measurement'

If you want to be freed from questions like, 'Physical quantities are operators, and they act on mysterious wave functions that are complex numbers? What is that wave function? Is it okay for it to collapse upon measurement, violating causality? Since it's not a continuous time evolution following the Schrödinger equation, isn't that collapse due to measurement strange?', then this textbook is recommended for you. The content is aimed at second- to third-year undergraduate physics students who have basic knowledge of mechanics, electromagnetism, analysis, and linear algebra.

'Introduction to Modern Quantum Mechanics' is a modern textbook for cultivating quantum natives. Following Victor Weisskopf, it is written with the spirit of "It doesn't matter what we cover in the textbook, it matters what you discover." The goal is not to cover standard themes like the hydrogen atom and explain them in an easy-to-digest way, but rather to pack the book with the fundamental ways of thinking for the new quantum era. 'Discover' contains the prefix 'dis-', which means negation. In other words, by removing the 'cover' (dis-cover), it literally means to discover. "It matters what you discover" means that what the reader newly discovers from this textbook is what is important. I have scattered clues for this not only in the main text but also in the notes and appendices. I hope you can feel the sense of realism in constructing quantum mechanical theory in the 21st-century style.

To describe the features of this textbook specifically, they are as follows.

(1) It emphasizes that quantum mechanics is a type of information theory and clarifies that the 'measurement problem' discussed in the last century is merely a pseudo-problem.

(2) A style of building theory using minimal experimental results. In particular, the Born rule of probability interpretation and quantum state superposition are not assumptions, but are derived. It provides a method to experimentally confirm whether an unknown system X placed in front of you is a genuine quantum system that satisfies the laws of quantum mechanics. In that task, there is no need to investigate all relationships with other physical systems in the universe; by experimenting only on X, you can reliably determine whether it is a quantum system or not. Textbooks that start by stating axioms and premises and only showing their consequences are mainstream, but they have significant disadvantages. No matter how many experiments you continue to perform to confirm the phenomena predicted by the axioms, it remains ambiguous at which stage of experimental reporting one can assert that the system is truly a quantum system. The weakness of textbooks starting from axioms is that they end up with a feeling of 'there are so many consistent experimental results, so it is probably a quantum system.'

Relativity is constructed starting from axioms that can be experimentally verified without combining other facts, such as the principle of relativity and the principle of the constancy of the speed of light. If experimental facts showing that the speed of light is not constant were to emerge, relativity itself could be refuted, but the axioms of quantum mechanics are different. For example, the commonly adopted axiom that 'the quantum state space is a complete Hilbert space' is a form that cannot be experimentally verified on its own, and as written in the following article, it is an assumption that can never be proven in the future.

Unlike textbooks in the axiomatic style that have such weaknesses, the style of this textbook has the strength that if you perform at least this much experimentation and it is consistent with the theory, you can assert that the system is a true quantum system that completely satisfies the laws of quantum mechanics.

Textbooks that start from an axiomatic system also have other problems. For example, by mathematically defining wave functions and state vectors as elements of a Hilbert space, they lead readers to think even later that 'the physical meaning of those wave functions and state vectors, including their reality, is unknown.' This leads to the tragedy of pushing readers into the swamp of a 'measurement problem' that does not even exist. This textbook completely blocks the entrance to that unnecessary maze.

(3) It provides a clear operational definition of physical quantities and quantum states (wave functions) in quantum mechanics. (In previous textbooks, the definition of the wave function was ambiguous or taken for granted.) For example, a high-precision quantum computer can be used as a world standard for the definition of physical quantities in the future.

(4) It clearly explains that matrices and complex vectors are merely notations for quantum mechanics, which is an information theory based on probability. This is the same as how Maxwell's equations of electromagnetism can be written in quaternion or exterior differential form notation. There is no inherent need to write quantum mechanics using matrices, complex vectors, or wave functions.

(5) It introduces basic items of quantum measurement theory and quantum information theory necessary for understanding quantum computers.

(6) Chapter 15 introduces a formulation that also handles general probability theory and information theory other than quantum mechanics, and adds a discussion on why nature chose the system called 'quantum mechanics' and implemented it in itself from among many such different possibilities.

(7) Appendix G provides a concrete example of a hidden variable theory describing a two-level spin system. Since this hidden variable theory can also be described by state vectors and density matrices, it clarifies that the Born rule using state linear superposition and projection operators is not a property unique to quantum mechanics, but also arises in certain types of classical hidden variable theories.

'Introduction to Modern Quantum Mechanics: Focusing on Quantum Information and Quantum Measurement' (Masahiro Hotta) | Kodansha BOOK Club (kodansha.co.jp)

[The Standard of This Century!]
For the next generation of physics students, I have fundamentally reconstructed quantum mechanics. 15 chapters to truly understand from the principles. A must-read book for undergraduates to experts.

[Table of Contents]
Chapter 1: Hidden Variable Theory and Quantum Mechanics
Chapter 2: Quantum Mechanics of Two-Level Systems
Chapter 3: Quantum Mechanics of Multi-Level Systems
Chapter 4: Quantum States of Composite Systems
Chapter 5: Correlations of Physical Quantities and Quantum Entanglement
Chapter 6: Quantum Operations and Time Evolution
Chapter 7: Quantum Measurement
Chapter 8: Quantum Mechanics of Particles in One-Dimensional Space
Chapter 9: Quantum Harmonic Oscillator
Chapter 10: Charged Particles in a Magnetic Field
Chapter 11: Quantum Behavior of Particles
Chapter 12: Spatial Rotation and Angular Momentum Operators
Chapter 13: Three-Dimensional Spherically Symmetric Potential Problems
Chapter 14: Quantum Information Physics
Chapter 15: Why Did Nature Choose 'Quantum Mechanics'?
Appendix

An electronic reference book by EMAN is also available, so please purchase and use it.

My book, 'Introduction to Modern Quantum Mechanics,' is also introduced in '100 Years of Quantum Mechanics' by Dr. Fumiaki Sato, who received the 78th Mainichi Publishing Culture Award.

It is also introduced in the Journal of the Physical Society of Japan as a textbook for an 'innovative period'.

I have prepared the following article as a supplementary guide for understanding when connecting from Chapter 2 to Chapter 3.

Note that there was criticism from someone in the Amazon comment section regarding the formulation of quantum mechanics in Chapter 3, but it is basically misguided criticism. Please refer to the following supplement regarding this.

As a supplementary reader to this 'Hotta Quantum Mechanics,' 'The New Common Sense of Quantum Mechanics: Physics Redefined by Consciousness, Reality, and Information' has also been published by Kodansha Scientific, co-authored with Katsuhiko Hiroe, who is famous for EMAN Physics.

The errata can be found below.
'Introduction to Modern Quantum Mechanics: Focusing on Quantum Information and Quantum Measurement' - Quantum Universe (hatenablog.com)