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College Physics

A demonstration of completeness of coordinate eigenvectors

LI Ti-jun

Published Jan 1, 2009• 0 citations
OpenAlex

Abstract

A vector |x〉expressed by the real number x,the creation operator and the ground state of particle number operator is obtained in terms of the eigenvectors and the completeness relation of particle number operator. It is proved that the vector |x〉is the eigenvector of coordinate operator;the integral over coordinate projective op- erators is equal to the identity operator;and the eigenvectors of coordinate operator are orthogonal.Hence all eigen- vectors {|x〉} form a complete and orthogonal set of basic vectors in an infinite-dimensional vector space.

Related papers

Expressions of coordinate eigen vectors and their integration over non-symmetric coordinate projective operators

LI Ti-jun, 菏泽学院物理系，菏泽 274015

The coordinate state vector 〈f(x)| is expressed with 〈x| in virtue of particle number states and the eigen equations of coordinate operator functions，accordingly we can convert non-symmetric projective operators into symmetric，the integrations over non-symmetric projective operators are obtained by the completeness relation of coordinate eigenstates.

Acta Physica Sinica•Jan 1, 2008•1 citation

Deriving the eigenvector of the operator λ■+μ■ by|x〉

Hongqi Li

By the representation of the coordinate eigenvector |x〉 in the Fock space,the eigenvector |ψ〉 of the operator λ■+μ■ is derived and its orthogonal relation is also proved.These results show that the eigenvector |ψ〉 satisfies the completeness and the orthogonal relations.Therefore,this eigenvector |ψ〉 can be qualified for a representation in quantum mechanics.When λ=1 and μ= 0,the eigenvector |ψ〉 evolves into the coordinate eigenvector |x〉;while λ=0 and μ =1,the |ψ〉 reduces to the momentum eigenvector |p〉.Therefore,the representation consisted by the eigenvectors |ψ〉 is an intermediate representation between the coordinate representation and the momentum representation.

College Physics•Jan 1, 2010•0 citations

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The Postulates of Quantum Mechanics

F. S. Levin

In this chapter the basic concepts of quantum theory are formulated via a self-consistent set of six postulates. These six postulates deal with the following: operator images of physical observables; properties of state vectors and wave functions; the connection with experimental measurements; coordinate-space forms of those operators having classical analogs; the primary dynamical equation of quantum mechanics; and the completeness of the eigenvectors of observables. The postulates are the framework on which the edifice of quantum mechanics is constructed. A separate section is devoted to the statement of each postulate and to a few examples chosen to illustrate or apply it. A comprehensive set of applications is presented in Chapters 6 and 7, where various 1-D systems are used as the vehicles for illustrating the postulates. The basic theory of this chapter is extended and developed in Chapters 8 and 9. Since probability is a key interpretational concept, a brief introduction to this topic is given in an appendix. We emphasize that our formulation of quantum theory (in this chapter and throughout this book) is restricted to the case of pure states. This is not a major restriction. It means that, when we consider ensembles of individual quantum systems (e.g., single particles, atoms, nuclei, etc.), all of the systems in the ensemble are assumed to be in the same quantum state, not necessarily an eigenstate of a quantal operator. Ensembles for which this assumption does not hold are described by a density operator or density matrix rather than by a pure state; treatments of such cases can be found in references cited later.

Cambridge University Press eBooks•Dec 3, 2001•0 citations

Differential Forms of Completeness Relation in Two-Mode 〈ξ|-〈η| Representations

Fan Hong-Yi, Liu NaiLe

On the basis of the common eigenvectors 〈η| and 〈ξ|, which respectively are eigenvectors of two particles' relative position and total momentum as well as center-of-mass coordinate and relative momentum, we find that there exist differential forms of completeness relation. The applications of the new completeness relation are pointed out. Throughout our discussion the technique of differentiation within an ordered product of operators is fully employed.

Communications in Theoretical Physics•Dec 15, 1999•0 citations
A vector |x〉expressed by the real number x,the creation operator and the ground state of particle number operator is obtained in terms of the eigenvectors and the completeness relation of particle number operator. It is proved that the vector |x〉is the eigenvector of coordinate operator;the integral over coordinate projective op- erators is equal to the identity operator;and the eigenvectors of coordinate operator are orthogonal.Hence all eigen- vectors {|x〉} form a complete and orthogonal set of basic vectors in an infinite-dimensional vector space.
By the representation of the coordinate eigenvector |x〉 in the Fock space,the eigenvector |ψ〉 of the operator λ■+μ■ is derived and its orthogonal relation is also proved.These results show that the eigenvector |ψ〉 satisfies the completeness and the orthogonal relations.Therefore,this eigenvector |ψ〉 can be qualified for a representation in quantum mechanics.When λ=1 and μ= 0,the eigenvector |ψ〉 evolves into the coordinate eigenvector |x〉;while λ=0 and μ =1,the |ψ〉 reduces to the momentum eigenvector |p〉.Therefore,the representation consisted by the eigenvectors |ψ〉 is an intermediate representation between the coordinate representation and the momentum representation.
We give the explicit form of the common eigenvectors of the relative position Q1{\mathit{Q}}_{1}Q1​-Q2{\mathit{Q}}_{2}Q2​ and the total momentum P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​, of two particles which were considered by Einstein, Podolsky, and Rosen [Phys. Rev. 47, 777 (1935)] in their argument that the quantum-mechanical state vector is not complete. Orthonormality and completeness of such eigenvectors, as well as their use in constructing the unitary operator for simultaneously squeezing Q1{\mathit{Q}}_{1}Q1​-Q2{\mathit{Q}}_{2}Q2​ and P1{\mathit{P}}_{1}P1​+P2{\mathit{P}}_{2}P2​, are derived by using the technique of integration within an ordered product of operators.
We construct 2N operators for a N-particle system, namely one center-of-mass coordinate operator, N-1 relative coordinate operators, one total momentum operator and N-1 mass-weighted relative momentum operators, and give common eigenvectors of N compatible observables [Formula: see text], which are composed of N particles' coordinate [Formula: see text] and momentum [Formula: see text]. By compatible, we mean such observables can be simultaneously determined. Using the technique of integration within an ordered product of operators (IWOP), we prove that the common eigenvectors are complete and orthonormal, and hereby qualified for making up a representation. This new representation can be applied to solving some dynamic problems in quantum mechanics.
In this chapter the basic concepts of quantum theory are formulated via a self-consistent set of six postulates. These six postulates deal with the following: operator images of physical observables; properties of state vectors and wave functions; the connection with experimental measurements; coordinate-space forms of those operators having classical analogs; the primary dynamical equation of quantum mechanics; and the completeness of the eigenvectors of observables. The postulates are the framework on which the edifice of quantum mechanics is constructed. A separate section is devoted to the statement of each postulate and to a few examples chosen to illustrate or apply it. A comprehensive set of applications is presented in Chapters 6 and 7, where various 1-D systems are used as the vehicles for illustrating the postulates. The basic theory of this chapter is extended and developed in Chapters 8 and 9. Since probability is a key interpretational concept, a brief introduction to this topic is given in an appendix. We emphasize that our formulation of quantum theory (in this chapter and throughout this book) is restricted to the case of pure states. This is not a major restriction. It means that, when we consider ensembles of individual quantum systems (e.g., single particles, atoms, nuclei, etc.), all of the systems in the ensemble are assumed to be in the same quantum state, not necessarily an eigenstate of a quantal operator. Ensembles for which this assumption does not hold are described by a density operator or density matrix rather than by a pure state; treatments of such cases can be found in references cited later.