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
5 - Hamiltonian 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
Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach.
The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space, defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian.
To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.
- Type
- Chapter
- Information
- Path Integrals and HamiltoniansPrinciples and Methods, pp. 80 - 104Publisher: Cambridge University PressPrint publication year: 2014