Concepts of states in physics

Aleksander Leonidovich Simanov

F. S. Levin

Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts. This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.

L. E. Ballentine

Ren ZHANG, Jinling Xie, Shengli Ma +2 more

Revisiting seemingly unrelated physical concepts from a unified perspective is not only a scientific research method but also aids in physics education, facilitating students' comprehensive understanding. Using a comparative analysis research method, this paper takes the SU(1,1) group as an example to discuss how to view several concepts in different fields of physics from a unified perspective, covering Lorentz transformations in special relativity, simple harmonic oscillators in quantum mechanics, Bogoliubov transformations in condensed matter physics, and squeezed states in quantum optics. By re-examining the examples involved in this paper from the perspective of the SU(1,1) group, on one hand, it deepens the understanding of the relevant concepts, and on the other hand, it helps to deepen students' understanding of the concept of symmetry. Revisiting several seemingly unrelated concepts in the field of physics from a unified perspective not only providesa new perspective for scientific research but also greatly promotes the depth and breadth of physics education. This paper takes the SU(1,1) group as an example to explore how to apply this mathematical tool to different fields of physics, thereby revealing its core role in different physical phenomena. The paper first analyzes the Lorentz transformation in special relativity, demonstrating how the SU(1,1) group helps us to understand the physical essence of this transformation more deeply. Then, the paper turns to quantum mechanics to discuss the application of the SU(1,1) group in describing the problem of the simple harmonic oscillator, revealing its key role in the evolution of quantum states. In the field of condensed matter physics, the paper discusses the unified description of Bogoliubov transformations, emphasizing the importance of the SU(1,1) group in understanding phase transitions in condensed matter systems. The paper also explores squeezed states in quantum optics, showing the application of the SU(1,1) group in describing the characteristics of light field squeezed states, further revealing its potential in quantum information processing. Through the perspective of the SU(1,1) group, this article summarizes the common mathematical foundations and universal physical laws of these physical concepts, and it not only deepens the understanding of these physical concepts but also promotes students' in-depth understanding of the concept of symmetry, thus providing a new perspective and method for physics education and research.

Nelson Enrique Bolivar, Gabriel Abellán

The uncertainty principle, formulated by Heisenberg in 1927, is a fundamental principle in quantum mechanics. It forces us to review key concepts of classical physics, and from this analysis are born the concepts of quantum physics. Consequences of the Uncertainty principle also introduces the concept of incompatible sizes, a completely new concept compared to classical physics. A state of the system, which we call a quantum state, will be determined from the knowledge of a set maximum of certain quantities that are independent and compatible with each other. A measurable physical quantity is called an observable. If at a given moment a measurement is performed by one or more compatible observables it is said they are performing an observation of the system. The basic principles of quantum mechanics are expressed by a number of postulates. Their numbers and can vary from one text to another.

L. D. Faddeev, O. Yakubovskiĭ