*Efstratios Manousakis*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198749349.003.0002
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter considers a two-state system in order to review the Dirac notation in the simplest realization of quantum mechanics, that is, a Hilbert space spanned by a basis of just two states.

This chapter considers a two-state system in order to review the Dirac notation in the simplest realization of quantum mechanics, that is, a Hilbert space spanned by a basis of just two states.

*Mike Finnis*

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198509776
- eISBN:
- 9780191709180
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509776.003.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter reviews the general concepts in quantum mechanics that are essential for the purpose of deriving models of interatomic forces in condensed matter. It assumes a familiarity with the usual ...
More

This chapter reviews the general concepts in quantum mechanics that are essential for the purpose of deriving models of interatomic forces in condensed matter. It assumes a familiarity with the usual material of a first course in quantum mechanics such as the Hamiltonian in operator notation, and the form of a time-independent Schrödinger equation. The chapter may serve as a guide to most of the notation used in the book, including Dirac notation. There are introductions to periodic boundary conditions, single particle Green functions, densities of states and pseudopotentials, which will be used later in the book.Less

This chapter reviews the general concepts in quantum mechanics that are essential for the purpose of deriving models of interatomic forces in condensed matter. It assumes a familiarity with the usual material of a first course in quantum mechanics such as the Hamiltonian in operator notation, and the form of a time-independent Schrödinger equation. The chapter may serve as a guide to most of the notation used in the book, including Dirac notation. There are introductions to periodic boundary conditions, single particle Green functions, densities of states and pseudopotentials, which will be used later in the book.

*Oliver Johns*

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

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics

Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions have a ...
More

Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. This chapter defines linear operators and discusses operators and matrices as well as special operators, dyadics, resolution of unity, complex vectors and operators, real and complex inner products, eigenvectors and eigenvalues, eigenvectors of real symmetric operator, eigenvectors of real anti-symmetric operator, normal operators, determinant and trace of normal operator, eigen-dyadic expansion of normal operator, functions of normal operators, exponential function, and Dirac notation.Less

Linear vector functions of vectors, and the related dyadic notation, are important in the study of rigid body motion and the covariant formulations of relativistic mechanics. These functions have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. This chapter defines linear operators and discusses operators and matrices as well as special operators, dyadics, resolution of unity, complex vectors and operators, real and complex inner products, eigenvectors and eigenvalues, eigenvectors of real symmetric operator, eigenvectors of real anti-symmetric operator, normal operators, determinant and trace of normal operator, eigen-dyadic expansion of normal operator, functions of normal operators, exponential function, and Dirac notation.

*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/9780191001628.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter introduces the concept of linear vector functions of vectors and the related dyadic notation, a concept that is particularly important in the study of rigid body motion and the covariant ...
More

This chapter introduces the concept of linear vector functions of vectors and the related dyadic notation, a concept that is particularly important in the study of rigid body motion and the covariant formulations of relativistic mechanics. Linear vector functions of vectors have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself, not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. The chapter begins with a definition of operators and matrices, and concludes with the Dirac Notation, used by quantum mechanics for complex vectors.Less

This chapter introduces the concept of linear vector functions of vectors and the related dyadic notation, a concept that is particularly important in the study of rigid body motion and the covariant formulations of relativistic mechanics. Linear vector functions of vectors have a rich structure, with up to nine independent parameters needed to characterise them, and vector outputs that need not even have the same directions as the vector inputs. The subject of linear vector operators merits a chapter to itself, not only for its importance in analytical mechanics, but also because study of it will help the reader to master the operator formalism of quantum mechanics. The chapter begins with a definition of operators and matrices, and concludes with the Dirac Notation, used by quantum mechanics for complex vectors.