Primitive spatial product

Although the linearity of the chain-rule differential expressions (10.5) confers primitive affine-type spatial structure on thermodynamic variables, it does not yet provide a sense of distance or metric on the space (other than what might be displayed in an arbitrarily chosen axis system). In order to bring intrinsic geometrical structure to the thermodynamic space, we need to define the scalar product (R RJ) [(9.29)] that dictates the spatial metric on Ms- The metric on Ms should reflect intrinsic physical properties of the thermodynamic responses, not merely generic chain rule-type mathematical properties of their differential representation. At the same time, we must exhibit how the space Ms is explicitly connected to the physical measurements of thermodynamic responses. Because such measurements assign scalar values to physical properties, it is natural to associate each scalar product of Ms with the scalar value of an experimental measurement. How can this be done ... 

We can obviously set up an heuristic primitive for the spatial part using the very idea which was used to generate Q/(. If each electron pair is a localised bond then the spatial part should be capable of being generated from a product of spatial orbitals similar to the primitive product used in the Heitler-London model of the electron-pair bond. Clearly, if these individual orbitals are formed from some set of basis functions, then the matrix which defines the orbitals in terms of the basis functions forms a familiar variational problem... 

The generalisation of the HL model applied to eqn (21.9) must therefore be the generalisation of the formation of the full spatial HL function from the primitive product the function Fk must simply be a sum of terms which can be generated from Q by permutations of the spatial variables of the electrons. That is... 

In principle, many atoms can be used for atomic-beam-holography purposes. It is possible to deposit the atoms directly on a substrate to produce a desired pattern, with a theoretical resolution of about 100 nm under typical conditions. Atomic-beam holography has considerable potential for the production of patterns with a nanometer-scale resolution. The present state of the art of atomic holography is rather primitive. However, it is a promising technique for atom manipulation in three-dimensional space, which could be used for control of the spatial phase and amplitude structure of atomic de Broglie waves in the future (Shimizu 2000). 

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