Symmetry and quantum mechanics (Q2832071)

The book under review is an introduction to applications of the theory of Lie groups in quantum mechanics. The text is structured as ``a conversation between a mathematician and a physicist. Starting with some basic physical intuitions and experimental results they set out to make a model of the Physical world.''NEWLINENEWLINEThe book consists in 9 chapters.NEWLINENEWLINEIn Chapters 1 and 2, the physical and the spinor spaces are introduced. The groups \(\mathrm{SO}(3)\) and \(\mathrm{SU}(2)\) are discussed.NEWLINENEWLINEIn Chapter 3, observable and the uncertainty principle are discussed. The Lie algebras \(su(2)\) and \(sl(2)_{\mathbb{C}}\) are discussed on the mathematical counterpart.NEWLINENEWLINEIn Chapter 4, the Schrödinger equation is introduced.NEWLINENEWLINEChapter 5 is devoted to higher spin. Representations of \(\mathrm{SU}(2)\) are classified.NEWLINENEWLINEIn Chapter 6, multiple particles are discussed, tensor products of \(\mathrm{SU}(2)\) representations are calculated.NEWLINENEWLINEIn Chapters 7 and 8, the observables of position and momentum are introduced. The Heisenberg group is discussed on the mathematical counterpart. In these chapters, some well-know physical models are discussed: the harmonic oscillator, the hydrogen atom and others.NEWLINENEWLINEFinally, in Chapter 9, the Galilean group and the Dirac equation is discussed.NEWLINENEWLINEThe author tries to avoid difficult questions form functional analysis underlying quantum mechanics. Thus, the explanation is simple, the book may be recommended to undergraduate students.NEWLINENEWLINEThe book may be recommended also to mathematicians as a rigorous introduction to applications of the Lie group in quantum mechanic. Since for each step a physical motivation is provided the book also may be interesting to physicists looking for short and rigorous introduction to mathematics underlying quantum mechanics.