Wave function in momentum space

The wave function in momentum space is given by the Fourier transform of the coordinate-space wave function... 

The function 4> k) is known as the wave function in momentum space. The Fourier integral represents the superposition of many waves of different wave vectors. This construct defines a wave packet, once considered as the theoretically most acceptable description of a wave-mechanical particle5. Schrodinger s dynamical equation (4) for a free particle... 

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. 

Here the Hamiltonian Ho describes the evolution of the polarization in an isolated IQW in the absence of electron-hole populations and corresponds to the Wannier equation (39). The resonant Wannier exciton wave function in momentum space Tk(q) is its eigenfunction with the eigenvalue Ew(k). The Hamiltonian H describes the nonlinear many-particle corrections. It is proportional... 

Thus, the adjoint relationship, expressed by the matrix G, is particularly simple. In quantum mechanics the coefficients ak have an important interpretation since they represent the amplitude of the wave function in momentum space. Equations (23) and (24) are direct analogues to the continuous Fourier transformation, which changes a coordinate... 

Figure 7 The four steps in the application of the kinetic energy operator by the Fourier method (A - B - C - D) (lower panel) coordinate space (upper panel) momentum space. A is the original wave function i i(g). B represents the wave function in momentum space 4>(p) obtained by Fourier transform of ty(q). C is the application of the kinetic energy operator in momentum space. (p) = p2/2M ij>(p) where T = p2/2M is also shown. D is (<7) = h is the final...

Figure 22 Wave packet propagation in a Coulomb experiment simulation of the process D - 2D". (Lower panel) Absolute value of the wave function in position space superimposed on the repulsive Coulomb potential. The wave functions are labeled by the elapsed time from the ionization in atomic units. (Upper panel) Absolute value of the wave function in momentum space superimposed on the kinetic energy term P2. Since the wave function is compact an adaptive grid which will follow the wave packet in position and momentum space can reduce the number of grid points significantly.

Electron Wave Functions in Momentum Space 9.2.1 Hydrogenic Wave Function... 

S.P. Alliluev [29,30] was able to obtain exact D-dimensional hydrogenlike wave functions in momentum space by a generalization of Fock s method. In Alliluev s treatment, Fock s transformation was generalized in such a way as to project D-dimensional momentum space onto the surface of a (DH-l)-dimensional hypersphere [24]. The momentum-space hydrogenlike wave functions could then be shown to be proportional to (D-f-l)-dimensional hyperspherical harmonics. 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

So A can act as a wave function and the Proca equation can be regarded as a quantum equation if A is a wave function in configuration space, and as a classical equation in momentum space. 

A much lesser known contribution of Pauling to the chemical knowledge, is his explicit expression for the momentum representation of the hydrogenic wave function [3]. Momentum space concepts are common among scattering physicists, some experimental chemists and a few theoreticians however, they have not won over the bulk of chemists nearly as efficiently as the hybrid concept. The reason is that they are somewhat counter intuitive and molecular structure is expressed in a rather indirect and (in the truest sense of the word) convoluted manner. 

In order to obtain the eigenfunctions of H y we have to apply the transformation operator U (Eq. (14)) to according to Eq. (13). This is a nontrivial task. The operator exp(ijSpiPy) acts on a function of the coordinates x and y and the result cannot be derived directly. It is very simple, though, to get the result if the operator acts on a function in momentum space. For the determination of the wave function we, therefore, proceed as follows. First we take the Fourier transform of the eigenfunction of H3. As a next step we apply the operator exp(ij8pjPy) which is in momentum space a simple multiplication. Next we take the inverse Fourier transform of the result in order to obtain the function / , j(x, y) in the coordinate space of H y. In the latter step we use the convolution theorem [13] for Fourier transforms. Subsequently applying the... 

If the r-space wave function is a linear combination of Slater determinants constructed from a set of spin-orbitals 0/, then its p-space counterpart is the same linear combination of Slater determinants constructed from the spin-momentals j obtained as Fourier transforms, Eq. (9), of the spin-orbitals. The overwhelming majority of contemporary r-space wave functions can be expressed as a linear combination of Slater determinants, and in these cases only three-dimensional Eourier transforms, Eq. (9), of the spin-orbitals are necessary to obtain the corresponding A -electron wave function in p space. Use of the full Eq. (5) becomes necessary only for wave functions, such as Hylleraas- or Jastrow-type wave functions, that are not built from a one-electron basis set. Examples of transformations to momentum space of such wave functions for He and H, can be found elsewhere [20-22]. 

Figure 5 (Upper panel) A wave packet, a semilocalized wave function in coordinate space (right panel) and in momentum space (left panel) where Pm = tt/Aq, L is the interval length L = NKAq). For the wave packet the convergence of the expansion (23) with respect to NK is exponential 0(e ). (Lower panel) A localized wave function in coordinate space but due to the sharp cutoff it is not localized in momentum space (left lower panel). Convergence with respect to NK is only 0(l/N ).

In the parabolic potential approximation there Is nonzero probability that the atom acquires a certain finite momentum, P, without change of the energy. This Is a purely quantum effect. The oscillator wave function In P space Is... 

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 symmetry properties of the momentum space wave functions can be obtained either from their position space counterparts or more directly from the counterpart of the Hamiltonian in momentum space. 

Show that the wave functions A (y) in momentum space corresponding to 0 ( ) in equation (4.40) for a linear harmonic oscillator are... 

Figure 5.1 illustrates the Fourier transform (FT) of a simple function, viz., a Gaussian. The relatively sharp Gaussian function with the exponent a = 1 depicted in Figure 5.1a, yields a diffuse Gaussian (in dotted line) in momentum space. A flat Gaussian function in position space with a = 0.1, transforms to a sharp one (cf. Figure 5.1b). Connected by an FT, the wave functions in position and momentum...

The atomic and molecular wave functions are usually described by a linear combination of either Gaussian-type orbitals (GTO) or Slater-type orbitals (STO). These expressions need to be multiplied by a center dependent factor expf ip-A). Further the STOs in momentum space need to be multiplied by Yim(6p,p). Examining the expressions [4], one notices the Gaussian nature of the GTOs even after the FT. The STOs are significantly altered on FT. From the expressions in Table 5.1, STOs are seen to exhibit a decay which is the decay of the slowest Is... 

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

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