Momentum Space Representation

So far only the position-space formulation of the (stationary) Dirac Eq. (6.7) has been discussed, where the momentum operator p acts as a derivative operator on the 4-spinor Y. However, for later convenience in the context of elimination and transformation techniques (chapters 11-12), the Dirac equation is now given in momentum-space representation. Of course, a momentum-space representation is the most suitable choice for the description of extended systems under periodic boundary conditions, but we will later see that it gains importance for unitarily transformed Dirac Hamiltonians in chapters 11 and 12. We have already encountered such a situation, namely when we discussed the square-root energy operator in Eq. (5.4), which cannot be evaluated if p takes the form of a differential operator. 

After Fourier transformation (see appendix E), the momentum-space representation of the Dirac equation is given by 

The solution of the Dirac equation does not need to be carried out explicitly as demonstrated for the position-space form in the beginning of this chapter. Instead the spinors are conveniently obtained by a Fourier transformation of the position-space solution functions. The Fourier transform of the spinor reads 

Schwabl provides many details on how the Dirac hydrogen atom can be solved, though some essential steps are treated very briefly. However, the material in Schw-abl s book is still more extensive than in many of the new publications on the subject. Also, the reader will find a more detailed comparison to the Klein-Gordon equation in external electromagnetic fields. But note that Schwabl chooses a different sign convention for the spherical spinors with effect on almost all equations of the Dirac hydrogen atom compared to Arose derived here. 

Salpeter, [81,135]. Quantum Mechanics of One- and Two-Electron Atoms. 

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Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

It is instructive to consider the momentum-space representation of the Gaussian wave packet. In this representation, the states are projected onto the eigenstates of the momentum operator, i.e., P p) = p p), which in the coordinate representation takes the form... 

The scattered state corresponding to xin) is Xout) = SlXin, and in the momentum-space representation... 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

Figure 5. Nodal surfaces of a sp d2z2 hybrid orbital with Z = 1 in the momentum-space representation. The left-hand plot contains two surfaces. One is the spherical node of the imaginary part. The second more complex surface consists of two closed and flattened spheres. These are the nodal surfaces belonging to the real part of the hybrid and are aligned along the -axis. The intersection of the two types of nodes are two circles around the -axis. The right-hand plot displays a cut through the a -plane. Note that the (polar) -axis is the horizontal axis in this plot. To avoid confusion, the nodal planes of the imaginary part are not displayed in either graph.

The momentum-space representation also proves particularly convenient for comparisons of the electron distributions of systems with different nuclear frameworks. Difference density plots in r-space are complicated by the different sets of nuclear positions. Such complications are absent in p-space and, in the case of polyenes [23], for example, momentum-space concepts have proved useful for examining the effects of bond alternation on the electron density - an important characteristic of such systems and of doped polyacetylene. 

Wavelets are a set of basis functions that are alternatives to the complex exponential functions of Fourier transforms which appear naturally in the momentum-space representation of quantum mechanics. Pure Fourier transforms suffer from the infinite scale applicable to sine and cosine functions. A desirable transform would allow for localization (within the bounds of the Heisenberg Uncertainty Principle). A common way to localize is to left-multiply the complex exponential function with a translatable Gaussian window , in order to obtain a better transform. However, it is not suitable when <1) varies rapidly. Therefore, an even better way is to multiply with a normalized translatable and dilatable window, v /yj,(x) = a vl/([x - b]/a), called the analysing function, where b is related to position and 1/a is related to the complex momentum. vl/(x) is the continuous wavelet mother function. The transform itself is now... 

In fact, in momentum space, the application of a nonlocal potential is numerically as easy as using the momentum-space representation of a local potential. 

McCoy and Sykes have introduced a method based on Fourier fransforming the ab initio wavefunction to generate a momentum space representation. Low momentum components of this function are then fitted to known values of the property under consideration for a series of molecules enabling the corresponding property of other molecules to be obtained by interpolation. 

Hence, one of the operators must be a differential operator while the other must be a simple multiplicative operator of the same variable. The first choice is called the position-space representation, while the second is called the momentum-space representation. Of course, one may add constants to these definitions but they are chosen to be zero since they would represent arbitrary shifts. Further, we must require that all arbitrary functions of position and momentum vanish. 

The connection to such a momentum-space representation as given above can also be made by starting from the cosine expression of the retarded interaction derived in the last section — through Eq. (8.35) — which finally produces [211]... 

For the discussion of the Douglas-Kroll-Hess transformation in chapter 12 it has been advantageous to consider the momentum-space representation of the Coulomb potential, which may be obtained via a Fourier transformation of V (r). It is given by... 

The set of coefficients ck constitutes the momentum space representation of function V -... 

To get the matrix elements of Vxc, we proceed as follow. First, we get the position space representation of p, by applying the FFT to equation (91). This will give us the value of the density at all the grid points, pj. Second, we compute the exchange-correlation potential v c at all the grid points, [vxcj]-Third, we apply the ITT to to get the momentum space representation of Vxc,... 

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