Momentum density Fourier transforms

The qualitative difference between low-density and high-density rotational relaxation is clearly reflected in the Fourier transform of the normalized angular momentum correlation function ... 

Mermin s "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin s reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space. 

The wave function W(x, i) may be represented as a Fourier integral, as shown in equation (2.7), with its Fourier transform A p, t) given by equation (2.8). The transform A p, i) is uniquely determined by F(x, t) and the wave function F(x, t) is uniquely determined by A p, i). Thus, knowledge of one of these functions is equivalent to knowledge of the other. Since the wave function F(x, /) completely describes the physical system that it represents, its Fourier transform A(p, t) also possesses that property. Either function may serve as a complete description of the state of the system. As a consequence, we may interpret the quantity A p, f)p as the probability density for the momentum at... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

The momentum expectation values (p ) are not directly related to the electron density, but to the wave function via its Fourier transform, the momentum density. However we can make use of a semiclassical relation for a local Fermi gas for estimating these values ... 

Note that the momentum density y(p) is not the Fourier transform of the position density p r). 

The r-space and p-space representations of the ( th-order density matrices, whether spin-traced or not, are related [127] by a fif -dimensional Fourier transform because the parent wavefunctions are related by a 3A -dimensional Fourier transform. Substitution of Eq. (5.1) in Eq. (5.8), and integration over the momentum variables, leads to the following explicit spin-traced relationship ... 

Since the momentum density is related to the reciprocal form factor or internally folded density by a Fourier transform, Eq. (5.29), there are sum rules that connect moments of momentum with the spherical average of B f) defined by... 

The two functions ip(q) and Fourier transforms of each other and they contain exactly the same information. By measuring the momentum density [Pg.231]

For a uniform charge distribution within a spherical atom the Fourier transform of the density has been shown (equation 5.6) to be of the form sin a/a, for a wave of phase a in momentum space. From the Bragg equation (Figure 2.9), A = 2dsin0, it follows that electrons at a distance d = A/2sin0 apart, scatter in phase, i.e. with phase difference 27T. At a separation r the relative phase shift a, is given by ... 

To obtain the momentum distribution 7ty(p) due to a single hybrid orbital ip(f), it is necessary to perform a Dirac-Fourier transform. The square magnitude of the resulting momentum orbital ip(p) is the contribution of ip to the momentum density ... 

Note, that the hybrid ip(r) is real. It may be written as a combination of an inversion-symmetric part ips and an antisymmetric part ipa. The Fourier transform will map the former onto the real part of ip, and the latter on its imaginary part, both of which are symmetric themselves. As a result, the square-magnitude of ip is inversion symmetric with respect to p = 0 (as momentum densities should be) [7],... 

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

Furthermore, the initial (ko) and final (ki) momenta of an impinging probe particle (neutron) may be assumed to be well defined [Squires 1996 van Hove 1954], Introducing the momentum transfer q = ko — ki from the probe particle to the scattering system, the Fourier transform of the particle density reads... 

The momentum-space electron density, p p), is just as straightforward to evaluate as its analogue in position spac, p(r). The wavefiinction in momentum space, (/ ), is simply the Fourier transform of the r-space wavefunction (r) ... 

The Fourier transform (Eq. (1)) preserves direction, in the sense that one can refer to components of the total momentum in any particular direction. For example, one can distinguish components along any Cartesian axis, or paral-lel/perpendicular to a bond or plane. A further consequence of Eq. (1) is that the momentum density p p) possesses the same symmetry elements as its r-space... 

Equation (41a) means that the function B( r) is equivalent to the volume integral of the density matrix y(ri, ri) under the condition of r = r - r, and Eq. (41b) means that B(r) is the autocorrelation function of the position wave function (r). The latter is an application of the Wiener-Khintchin theorem (Jennison, 1961 Bracewell, 1965 Champeney, 1973), which states that the Fourier transform of the power spectrum is equal to the autocorrelation function of a function. Equation (41c) implies not only that B(r) is simply the overlap integral of a wave function with itself separated by the distance r (Thulstrup, 1976 Weyrich et al., 1979), but also that the momentum density p(p) and the overlap integral S(r) are a pair of the Fourier transform. The one-dimensional distribution along the z axis, B(0, 0, z), for example, satisfies... 

Although the wavefunctions (pi,- , piv) and M(n, , rN) are connected by a Fourier transformation, their corresponding momentum and position one-particle densities ir (p) and pf(r), respectively, are not. This fact implies, in turn, that intra-orbit optimizations in position and momentum spaces are not equivalent. 

It is clear from Eqs. (18) and (19) that the number and momentum densities are not related by Fourier transformation. This is most readily understood for a one-electron system where the r-space density is just the squared magnitude of the orbital and the p-space density is the squared magnitude of the Fourier transform of the orbital. The densities are not Fourier transforms of one another because the operations of Fourier transformation and taking the absolute value squared do not commute. Moreover, there is no known direct and practical route from one density to the other even though the Hohenberg-Kohn theorem [32] guarantees that it must be possible to obtain the ground state n(p) from p(r). 

What does Fourier transformation of the momentum density yield This question has been considered [29] in some generality and detail. Here we merely summarize the outcome for the one-electron momentum density [28,29]. Consider the Fourier transform, or characteristic function in the terminology of probability theory, of 7T(p) ... 

There are two main methods for the reconstruction of 7T(p) from the directional Compton profile. In the Fourier-Hankel method [33,51], a spherical harmonic expansion of the directional Compton profile is inverted term-by-term to obtain the corresponding expansion of /T(p). In the Fourier reconstraction method [33,34], the reciprocal form factor B0) is constructed a ray at a time by Fourier transformation of the measured J(q) along that same direction. Then the electron momentum density is obtained from B( ) by using the inverse of Eq. (22). A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

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