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Macmillan Higher Education Palgrave Higher Education

Quantum Theory

A Mathematical Approach

ISBN 9783319376028
Publication Date September 2016
Formats Paperback Ebook Hardcover 
Publisher Springer

This book was inspired by the general observation that the great theories of modern physics are based on simple and transparent underlying mathematical structures – a fact not usually emphasized in standard physics textbooks – which makes it easy for mathematicians to understand their basic features. 

It is a textbook on quantum theory intended for advanced undergraduate or graduate students: mathematics students interested in modern physics, and physics students who are interested in the mathematical background of physics and are dissatisfied with the level of rigor in standard physics courses. More generally, it offers a valuable resource for all mathematicians interested in modern physics, and all physicists looking for a higher degree of mathematical precision with regard to the basic concepts in their field.

Peter Bongaarts taught for many years theoretical and mathematical physics at the University of Leiden, the Netherlands

Introductory remarks
Historical background
Physics in The Twentieth Century and Beyond
Methodological Remarks
Classical Mechanics
Introduction.-- Historical Remarks
Newtonian Classical Mechanics
The Lagrangian Formulation of Classical Mechanics
The Hamiltonian Formulation of Classical Mechanics
An Intrinsic Formulation
An Algebraic Reformulation
Quantum Theory: General Principles
Historical Background
The Beginning of Quantum Mechanics
Quantum Theory. General Remarks
The basic concepts of quantum theory
A preview
States and Observables
Time Evolution
Quantum Mechanics of a Single Particle I
`Diagonalizing' the Pj . The Fourier Transform
A General Uncertainty Relation
The Heisenberg Uncertainty Relation
Minimal Uncertainty States
The Heisenberg Inequality. Examples
The 3-dimensional Case.5
Quantum Mechanics of a Single Particle II
Time Evolution of Wave Functions
Pseudo-Classical Behavior of Expectation Values
The Free Particle
A Particle in a Box
The Tunnel Effect
The Harmonic Oscillator
The Classical Harmonic Oscillator
The Quantum Oscillator
Lowering and Raising Operators
Time Evolution
Coherent States
Time Evolution of Coherent States
The 3-Dimensional Harmonic Oscillator
The Hydrogen Atom
Many-Particle Systems
Combining Quantum Systems - Systems of N Particles
System of Identical Particles
An Example: The Helium Atom
Historical Remarks
The Fock Space Formulation for Many-Particle Systems
A Heuristic Formulation. `Second Quantization'
Review of Classical Statistical Physics
Classical Statistical Physics (Continued)
The Three main Ensembles
The Microcanonical Ensemble
The Canonical Ensemble
The Grand Canonical Ensemble
The Canonical Ensemble in the Approach of Gibbs
From Statistical Mechanics to Thermodynamics
Kinetic Gas Theory
General Statistical Physics
Quantum Statistical Physics
What is an Ensemble in Quantum Statistical Physics?
An Intermezzo - Is there a Quantum Phase Space?
An Approach in Terms of Linear Functionals
An Extended System of Axioms for Quantum Theory
The Explicit Form of the Main Quantum Ensembles
Planck's Formula for Black Body Radiation
Bose-Einstein Condensation
Physical Theories as Algebraic Systems
`Spaces'. Commutative and Noncommutative
An Explicit Description of Physical Systems I
An Explicit Description of Physical Systems II
Quantum Theory: Von Neumann Versus Birkhoff
Quantization. The Classical Limit
Towards Relativistic Quantum Theory
Einstein's Special Theory of Relativity
The Klein-Gordon Equation
The Dirac Equation
An Introduction to Quantum Field Theory
Introductory Remarks. Some History
Quantum Field Theory as a Many Particle Theory
Fock Space and its Operators
The Scalar Quantum Field
The Scalar Quantum Field
The Field Operators
The Scalar Field with Self-Interaction
Towards a Rigorous Quantum Field Theory
Concluding Remark


“This is a text intended primarily for advanced undergraduate and graduate students in mathematics, and secondarily for physiscs students wanting an introduction to the modern mathematics utilized in quantum mechanics. It has an intentional algebraic flavor that provides a succinct means for comparison between quantum and classical machanics. … The reader could infer that quantum theory, with its axioms, is tidy and has a well-defined mathematical framework.” (Lawrence E. Thomas, Mathematical Reviews, July, 2015)“This publication provides a quite interesting text book on quantum theory written having in mind advanced undergraduate or graduate students in mathematics, but which can also be a very nice reference text for physics students having an interest in the mathematical foundations of quantum theory.” (Bassano Vacchini, zbMATH 1315.81001, 2015)
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