Mathematical Formulation of Quantum Systems

Masahito Hayashi

Walter E Thirring

CARLOS DESA

This book presents a concise introduction to the mathematical structure of quantum mechanics from the perspective of operator algebras for systems with a finite number of degrees of freedom. It begins with a review of the mathematical tools required throughout the text, followed by an exposition of the standard Schrödinger formulation based on the Dirac–von Neumann axioms. The operator algebraic formulation is then developed from a set of physically motivated principles, providing an alternative and conceptually transparent description of quantum systems. The book concludes with an application comparing the standard and operator algebraic formalisms. Written in Spanish, this book is intended primarily for Spanish-speaking undergraduate and graduate students in physics, mathematics, and mathematical physics, aiming to provide accessible access to modern mathematical formulations of quantum mechanics.

K. Kong Wan

A physical theory often admits different mathematical descriptions. Classical mechanics has three familiar formulations: the traditional Newton’s formulation, the Lagrange’s and the Hamilton’s formulations. It is easy to see that these are physically equivalent. Quantum mechanics also admits many different formulations. This chapter discusses how different mathematical descriptions of a probability theory based on a Hilbert space can be physically equivalent. The prescription does not single out a unique Hilbert space nor a unique set of unit vectors and operators for the description of a given quantum system. This non-uniqueness does not matter since operators and unit vectors themselves are not physically measurable quantities. Two different mathematical descriptions of a quantum system are physically equivalent if the unit vectors and selfadjoint operators for the representation of states and observables in the two descriptions are related by a simultaneous unitary transformation.

Mou-Hsiung Chang