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Journal of Physics A Mathematical and General

Boson representations of one-dimensional scattering

Toru Miyazawa

Published Dec 15, 1999• 8 citations
Publisher

Abstract

A new description of one-dimensional scattering processes in terms of boson operators is presented, and the Schrödinger equation in a general form is analysed in this description on the basis of the factorization scheme. As an example of application, a low-energy expansion formula of the Green function is derived within the framework of this formalism.

DOI
10.1088/0305-4470/33/1/312

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A new description of one-dimensional scattering processes in terms of boson operators is presented, and the Schrödinger equation in a general form is analysed in this description on the basis of the factorization scheme. As an example of application, a low-energy expansion formula of the Green function is derived within the framework of this formalism.
The traditional basis of description of many-particle systems in terms of Green functions is here generalized to the case when the system is nonextensive, by incorporating the Tsallis form of the density matrix indexed by a nonextensive parameter q. This is accomplished by expressing the many-particle q Green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical Green function which now depends on this parameter. We study one- and two-particle Green functions in detail. From the one-particle Green function, we deduce some experimentally observable quantities such as the one-particle momentum distribution function and the one-particle energy distribution function. Special forms of the two-particle Green functions are related to physical dynamical structure factors, some of which are studied here. We deduce different forms of sum rules in the q formalism. A diagrammatic representation of the q Green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar. Approximation schemes for one-particle q Green functions such as Hartree and Hartree-Fock schemes are given as examples. This extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages.
The functional formalism, providing a unified treatment of the optical coherence theory is developed for arbitrarily ordered field operators. Multimode ordered expansions of operators in terms of boson amplitude operators as well as integral expansions are discussed. They form a basis of associating a class of s-ordered probability functionals with the density operator of the field. As an application, the distributions of photons and photoelectrons are studied. The generalized photo-detection equation of Perina and Horak (1970) is rederived by using the functional approach. A formula expressing the photocount generating function through the characteristic functional of the field is derived.
We present a new derivation of Time-Independent Perturbation Theory (PT) that is shorter and more general than the usual ones presented in Quantum Mechanics (QM) texts. Expressing the task of PT in terms of Green's functions, the key to our approach is to write the bound-state equation for the perturbed wave function directly in terms of the unperturbed Green's function. Not only does this greatly simplify the derivation of PT expressions for energies and wave functions, but it does so without using the orthogonality or completeness properties of unperturbed Hamiltonian eigenstates, which form the basis of standard derivations of PT. As a result, our approach does not require the potentials to be energy independent as is required in standard derivations, nor does it require the inverse free Green's operator G0−1(E) to be a linear function of energy, as is the case in QM. Consequently, our approach is applicable not only to QM but also to various extensions of QM, including Quantum Field Theory. For students, our approach provides an easy-to-understand introduction to the use of Green's functions in QM while at the same time presenting a description of PT that is shorter and more fundamental than the standard ones currently advocated.