1. Vectors and Operators

S. D. Deshpande, R. B. Pode

The bases used in quantum mechanics, in general, are orthonormal. The relations in orthonormal basis vectors are simpler than those in nonorthogonal basis vectors. Also, the operators that represent different physical quantities are Hermitian and the eigenvectors of these operators belonging to different eigenvalues are orthonormal to each other. But, in many cases, such as sets of screened hydrogenic wavefunctions, nonorthogonal sets of basis vectors occur. It is, therefore, necessary to consider the representation of vectors and operators in nonorthogonal bases. Making use of the set of vectors reciprocal to the given set of nonorthogonal base vectors, the expression for the norms of vectors and for the matrix elements has been found. Also, adjoint and Hermitian matrices in nonorthogonal bases have been worked out using a set of reciprocal basis vectors.

LI Ti-jun

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.

Joseph M. Renes

Oliver Davis Johns

Abstract 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.