*Laura Ruetsche*

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199535408
- eISBN:
- 9780191728525
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535408.003.0002
- Subject:
- Philosophy, Philosophy of Science

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter ...
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This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.Less

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.

*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 ...
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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.