Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 1 Synopsis
- Part one Fundamental principles
- 2 The mathematical structure of quantum mechanics
- 3 Operators
- 4 The Feynman path integral
- 5 Hamiltonian mechanics
- 6 Path integral quantization
- Part two Stochastic processes
- Part three discrete degrees of freedom
- Part four Quadratic path integrals
- Part five Action with acceleration
- Part six Nonlinear path integrals
- References
- Index
2 - The mathematical structure of quantum mechanics
from Part one - Fundamental principles
Published online by Cambridge University Press: 05 April 2014
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- 1 Synopsis
- Part one Fundamental principles
- 2 The mathematical structure of quantum mechanics
- 3 Operators
- 4 The Feynman path integral
- 5 Hamiltonian mechanics
- 6 Path integral quantization
- Part two Stochastic processes
- Part three discrete degrees of freedom
- Part four Quadratic path integrals
- Part five Action with acceleration
- Part six Nonlinear path integrals
- References
- Index
Summary
An examination of the postulates of quantum mechanics reveals a number of fundamental mathematical constructs that form its theoretical underpinnings. Many of the results that are summarized in this Chapter will only become clear after reading the rest of the book and a re-reading may be in order.
The dynamical variables of classical mechanics are superseded by the quantum degree of freedom. An exhaustive and complete description of the indeterminate degree of freedom is given by its state function, which is an element of a state space that, in general, is an infinite-dimensional linear vector space. The properties of the indeterminate degree of freedom are extracted from its state vector by the linear action of operators representing experimentally observable quantities. Repeated applications of the operators on the state function yield the average value of the operator for the state [Baaquie (2013e)].
The conceptual framework of quantum mechanics is discussed in Section 2.1. The concepts of degree of freedom, state space and operators are briefly reviewed in Sections 2.3–2.5. Three distinct formulations of quantum mechanics emerge from the superstructure of quantum mechanics and these are briefly summarized in Sections 2.7–2.9.
The Copenhagen quantum postulate
The Copenhagen interpretation of quantum mechanics, pioneered by Niels Bohr and Werner Heisenberg, provides a conceptual framework for the interpretation of the mathematical constructs of quantum mechanics and is the standard interpretation that is followed by the majority of practicing physicists [Stapp (1963), Dirac (1999)].
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- Chapter
- Information
- Path Integrals and HamiltoniansPrinciples and Methods, pp. 7 - 29Publisher: Cambridge University PressPrint publication year: 2014