Read more
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.
List of contents
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 $${hat{L}}$$z-adapted graphs.- 1.5 ($${hat{L}}$$z,?z)-adapted graphs.- 1.6 ?2 -adapted graphs.- 1.7 ($${hat{L}}$$z,?2)-adapted graphs.- 1.8 ($${hat{L}}$$2,?2)-adapted graphs.- 1.9 (?2,$${hat{T}}$$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.- 2.2 General formulas for matrix elements.- 2.3 Matrix elements in the ?z and $${hat{L}}$$z- adapted spaces.- 2.4 Reduction from ?z to ?2 eigenspace.- 2.5 Matrix elements in the ?2-adapted space.- 2.6 Non-fagot graphs and the ?2-adapted space.- References.
