Charge and Momentum Density

The electron density in an atom, molecule or crystal is described by a wave function which is subject to strict characteristic boundary conditions. As shown before (eqn. 5.3) an electron on a spherically symmetrical atom obeys the one-dimensional radial Helmholtz equation 

Likewise, an equivalent wave function is defined in momentum space 

The two functions ip(q) and p(k) are known as Fourier transforms of each other and they contain exactly the same information. By measuring the momentum density p2 it is therefore possible to determine the charge density 

Like the momentum variable, which in coordinate space is represented by 

We note that the quantum-mechanical state of a photon, the counterpart of a particle in atomic systems, is described by a wave function in momentum space [15]—p.246. Electromagnetic waves, such as X-rays, that are scattered on an electron, are of this type. Taking the Fourier transform of such a scattered wave must therefore reveal the position of the scatterer. 

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Becker P, Gillet J-M, Cortona P, Ragot S (2001) Complementary aspects of charge and momentum densities for the study of the chemical bond. Theor Chem Acc 105 284-291... 

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