Representations of Vectors and Operators

Sadri Hassani

Masashi Ban

Phase-space representation of quantum state vectors has been recently formulated by means of the relative-state method developed by the present author [J. Math. Phys. 39, 1744 (1998)]. It is, however, pointed out by Mo/ller that the displacement-operator method provides another basis of phase-space representation of quantum state vectors [J. Math. Phys. (to appear)]. Hence the relation between the relative-state approach and the displacement-operator approach is discussed, both of which yield equivalent phase-space representations.

Klaus B. Mo ller

A simple approach to phase-space representation of quantum state vectors using the displacement-operator formalism is presented. Although the resulting expressions for the fundamental operators (position and momentum) are equivalent to those obtained by other methods, this approach provides both alternative mathematical foundation as well as physical interpretation of phase-space representation of quantum state vectors.

Brian L. Silver

Boris Leaf

A basis of products of one-particle vectors (symmetric or antisymmetric for bosons or fermions) is defined, and related to the occupation-number basis in an abstract Fock space, a generalization of the wave-mechanical configuration representation of the space of second quantization. Creation and destruction operators are simply defined in the product-vector basis and shown to have their usual properties in occupation-number representation. Algebraic properties of product vectors are developed and illustrated. The representation of operators in the product-vector basis is described, together with a brief discussion of the effects on these operators of transformations from one set of one-particle vectors to another. A simple treatment of density and reduced density operators for systems of bosons or fermions is given in the product-vector basis. The relationship between degeneracy in the Fock-space spectrum of an additive operator NA1 and degeneracy in the spectrum of one-particle operator A1 is obtained.

George C. Turrell