*Oliver Johns*

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.001.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A ...
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This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.Less

This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.

*Oliver Davis Johns*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780191001628
- eISBN:
- 9780191775161
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780191001628.003.0014
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the ...
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This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the Klein-Gordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phase-space variables.Less

This chapter uses the traditional Hamilton equations as the basis for an extended Hamiltonian theory in which time is treated as a coordinate. The traditional Hamilton equations, including the Hamiltonian form of the generalised energy theorem, will be combined into one set of extended Hamilton equations. The extended Hamilton theory developed in the chapter is of fundamental importance for the more advanced topics in mechanics. It is used to write the relativistically covariant Hamiltonian, which is then used to derive the Klein-Gordon equation of relativistic quantum mechanics. The extended Hamilton equations also provide the basis for the discussion of canonical transformations. The objective of extended Hamiltonian theory is to write the equations of motion in terms of an extended set of phase-space variables.

*Efstratios Manousakis*

- Published in print:
- 2015
- Published Online:
- December 2015
- ISBN:
- 9780198749349
- eISBN:
- 9780191813474
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198749349.001.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The book contains lectures notes for a graduate two-semester course in quantum mechanics. It differs from other quantum mechanics textbooks as various parts of the book are inspired by rather recent ...
More

The book contains lectures notes for a graduate two-semester course in quantum mechanics. It differs from other quantum mechanics textbooks as various parts of the book are inspired by rather recent advances in various areas of physics. For example, the book begins by putting the Schrödinger equation on a spatial discrete lattice, inspired by Hamiltonian lattice gauge theories (HLGT). The book also discusses the path integral formulation of quantum mechanics and emphasize the adiabatic time evolution in the case of a time-dependent Hamiltonian. As an example of how to use symmetry in quantum mechanics, the book treats one-dimensional periodic potentials. The book also discusses atoms and molecules using mean-field-like treatment, such as the Hartree–Fock approximation, including a discussion on how to go beyond it. Electron–electron correlations in the hydrogen molecule are taken into account with a first quantized formulation of the two-site Hubbard model, which is solved analytically. The book also uses the canonical Hamiltonian quantization of quantum electrodynamics after finding the normal modes, in an analogy with the treatment of the normal modes of an array of atoms, the photons emerge as the quanta of such normal modes, in the same way as the phonons emerge in the treatment of the normal modes of the coupled array of atoms. This Hamiltonian quantization of the electromagnetic field is used later to treat its interaction with atomic matter, without having to follow the usual semiclassical treatment.Less

The book contains lectures notes for a graduate two-semester course in quantum mechanics. It differs from other quantum mechanics textbooks as various parts of the book are inspired by rather recent advances in various areas of physics. For example, the book begins by putting the Schrödinger equation on a spatial discrete lattice, inspired by Hamiltonian lattice gauge theories (HLGT). The book also discusses the path integral formulation of quantum mechanics and emphasize the adiabatic time evolution in the case of a time-dependent Hamiltonian. As an example of how to use symmetry in quantum mechanics, the book treats one-dimensional periodic potentials. The book also discusses atoms and molecules using mean-field-like treatment, such as the Hartree–Fock approximation, including a discussion on how to go beyond it. Electron–electron correlations in the hydrogen molecule are taken into account with a first quantized formulation of the two-site Hubbard model, which is solved analytically. The book also uses the canonical Hamiltonian quantization of quantum electrodynamics after finding the normal modes, in an analogy with the treatment of the normal modes of an array of atoms, the photons emerge as the quanta of such normal modes, in the same way as the phonons emerge in the treatment of the normal modes of the coupled array of atoms. This Hamiltonian quantization of the electromagnetic field is used later to treat its interaction with atomic matter, without having to follow the usual semiclassical treatment.