Grms or Graphical Representation of Model Spaces: Vol. 1 Basics

ISBN-13: 9783540171690 / Angielski / Miękka / 1986 / 189 str.

Wlodzislaw Duch

Opis

The purpose of these notes is to give some simple tools and pictures to physicists and ' chemists working on the many-body problem. Abstract thinking and seeing have much in common - we say "I see" meaning "I understand" , for example. Most of us prefer to have a picture of an abstract object. The remarkable popularity of the Feynman diagrams, and other diagrammatic approaches to many-body problem derived thereof, may be partially due to this preference. Yet, paradoxically, the concept of a linear space, as fundamental to quantum physics as it is, has never been cast in a graphical form. We know that is a high-order contribution to a two-particle scattering process (this one invented by Cvitanovic(1984)) corresponding to a complicated matrix element. The lines in such diagrams are labeled by indices of single-particle states. When things get complicated at this level it should be good to take a global view from the perspective of the whole many-particle space. But how to visualize the space of all many-particle states ? Methods of such visualization or graphical representation of the ,spaces of interest to physicists and chemists are the main topic of this work.

Spis treści

1. Preface.- 2. Introduction.- I: Architecture of Model Spaces.- 1.1 Introducing graphical representation.- 1.2 Labeling and ordering the paths.- 1.3 ? z-adapted graphs in different forms.- 1.4 $${\hat}$$z-adapted graphs.- 1.5 ($${\hat}$$z,?z)-adapted graphs.- 1.6 ?2 -adapted graphs.- 1.7 ($${\hat}$$z,?2)-adapted graphs.- 1.8 ($${\hat}$$2,?2)-adapted graphs.- 1.9 (?2,$${\hat}$$2)-adapted graphs.- 1.10 Spatial symmetry in the graph.- 1.11 Visualization of restricted model spaces.- 1.12 Physical intuitions and graphs.- 1.13 Mathematical remarks.- 1.14 Graphs and computers.- 1.15 Summary and open problems.- II: Quantum Mechanics in Finite Dimensional Spaces.- 2 Matrix elements in model spaces.- 2.1 The shift operators.- A. Definitions.- B. Properties of the shift operators.- C. Examples of operators in Êij basis.- 2.2 General formulas for matrix elements.- 2.3 Matrix elements in the ?z and $${\hat}$$z- adapted spaces.- A. The three-slope graphs.- B. Classification of loops in the three-slope graphs.- C. Graphical rules for matrix elements.- D. Example.- E. Four-slope graphs.- F. Other non-fagot graphs.- G. Matrix elements in the $${\hat}$$z and ($${\hat}$$z, ?z)-adapted spaces.- 2.4 Reduction from ?z to ?2 eigenspace.- 2.5 Matrix elements in the ?2-adapted space.- A. Permutations in the spin space.- B. Spin function transformation (SFT) graph and table.- C. Manipulations with permutations.- D. Presence of the singlet-coupled pairs.- E. Products of shift operators.- F. Evaluation of matrix elements in the ($${\hat}$$z,?2) eigenspace.- 2.6 Non-fagot graphs and the ?2-adapted space.- A. One-body segments.- B. Two-body segments.- C. Summary.- References.