From spinors to quantum mechanics / Gerrit Coddens (Commissariat à l'énergie atomique et aux énergies alternatives, France).

Format:

Book

Author/Creator:

Coddens, Gerrit, author.

Language:

English

Subjects (All):

Group theory.

Spinor analysis.

Quantum theory.

Physical Description:

xix, 383 pages : illustrations ; 24 cm

Place of Publication:

London : Imperial College Press, [2015]

Summary:

Quantum mechanics is a notoriously difficult subjects and is well-known to be highly counter-intuitive. In many cases, students are told to 'shut up and calculate', without being encouraged to understand fully the reasoning behind the calculations being made. With the aim to improve on this situation, From Spinors to Quantum Mechanics takes the unique approach of using geometry to understand quantum mechanics. This is presented in two parts, one mathematical and one physical. The former introduces group theory with unprecedented clarity, and provides an exact geometrical intuition of what a spinor is. The section is based on a balance between geometrical intuition and algebraic rigor, and provides an in-depth understanding of the subject matter to ensure that group-theoretical calculations no longer appear abstract or intimidating. The physical part builds on this and elucidates the new approach to studying the foundations of quantum mechanics using the established geometrical meaning of spinors. The development starts with a mathematically rigorous derivation of the Dirae equation, based on the assumption that electron is a spinning particle. Building the approach on the Dirac equation permits us to see the full symmetry of the theory, which otherwise could become concealed by introducing approximations. This derivation then serves as a starting point for a study of a number of standard topics of the traditional quantum mechanics course, like the hydrogen atom, the Zeeman effect and the double-slit experiment. In common approaches to quantum mechanics, where the geometrical meaning of algebra is ignored, it is impossible to delineate or even imagine the geometrical limits imposed on the algebra by the assumptions needed to construct a classical theory. In unawareness of these limits, however, one will feel free to cross unwittingly the frontiers between classical and quantum mechanics, and ultimately stretch the algebra beyond its initially conceived domain of definition. The meaning of quantum mechanics resides forcedly in the geometrical meaning of these "forbidden" extension, and it can be determined very precisely where and how they take place. The book will be of great value to all mathematicians and physicists who desire to fully understand the conceptual ideas behind the representation theory of the rotations and Lorentz groups. Physicists with an interest in the meaning and foundations of quantum mechanics will discover an unsuspected geometrical aspect that they just cannot afford to dispense with. Book jacket.

Contents:

Introduction to groups

Spinors in the rotation group

Spinors in the homogeneous Lorentz group

The Dirac equation from scratch

Towards a better understanding of quantum mechanics

The hidden variables issue and the Bell inequalities

Equivalence of the Bohr-Sommerfeld and Dirac theories for the hydrogen atom

The problem of the electron spin within a magnetic field

The double-slit experiment and the superposition principle

A caveat about the limitations of group theory

Spin and angular momentum as vector and bi-vector concepts.

Notes:

Includes bibliographical references (pages 371-373) and index.

ISBN:

9781783266364

1783266368

OCLC:

898162061