Quantum mechanics in matrix form
In a series of papers from 1925, Heisenberg took a decisive step in loosing the last tights with the image of classical motion inside atoms. By giving up the description of motion via a time-dependent coordinate \(x(t)\), he introduced instead the concept of quantum amplitude. The scope of the book under review is to present at an elementary level the method developed by Heisenberg. The book begins by exposing the quantum theory before 1925, as introduced by Bohr and Sommerfeld: quantization rule, Bohr's postulates and atom sizes. Chapter 2 presents the main ideas of Heisenberg as put forward by him in 1925, i.e, the description of quantum motion via transition amplitudes \(x_{mn}\) arranged in a matrix instead of \(x(t)\). The further developments of the matrix method, operated by Born, Heisenberg and Jordan are exposed in the next chapter (note that the reference that to the date provides the most comprehensive presentation of matrix quantum mechanics is the ``Elementare Quantenmechanik'' by Born and Jordan). Chapter 4 introduces the concepts of state vectors, density matrices for the description of mixed states and the uncertainty relations of Heisenberg are derived and interpreted. Chapters 5 to 10 are dedicated to the most notorious applications of the matrix method: One-dimensional harmonic oscillator, three-dimensional rotator, hydrogen atom, spin-orbit coupling, atom in electromagnetic field (normal and anomalous Zeeman effect) and fermion and boson many-particle systems. Chapter 11 performs a necessary step: the comparison of Heisenberg (matrix) and Schrödinger (wave) mechanics. The last chapter deals with the incorporation of special relativity to quantum mechanics, i.e. provides a simplified exposition of Dirac's relativistic electron theory from Chapter XI of the fourth edition of his book ``The principles of quantum mechanics''. Each chapter is endowed with problems, and the book can be an excellent companion to the standard undergraduate course of quantum mechanics.
scientific article; zbMATH DE number 6518659