Momentum space wave functions

The transformations connecting the coordinate-space wave function, v[/(R), to the momentum-space wave function, k), are... 

The discretized momentum-space wave function corresponding to a momentum of ki% is denoted by 44. As with the discretized spatial wave function [Eq. (37)], the discretized momentum wave functions are also normalized so that 4/ p = 1 (i.e., = i/ ki) V ). 

The discretized momentum space wave function, is therefore given by... 

In the next setion we review some key concepts in Mermin s approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV. 

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. 

E. Weigold, Momentum Space Wave Functions. American Institute of Physics, vol. 86, Adelaide, 1982... 

The transform A(p, t) is ealled the momentum-space wave function, while (jc, /) is more accurately known as the coordinate-space wave function. When there is no confusion, however, (jc, /) is usually simply referred to as the wave function. 

The expectation value p) of the momentum p may be obtained using the momentum-space wave function A p, i) in the same way that (x) was obtained from F(x, i). The appropriate expression is... 

What is the probability density as a function of the momentum p of an oscillating particle in its ground state in a parabolic potential well (First find the momentum-space wave function.)... 

The many-particle momentum space wave function, P2, P3,..., P/v) 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. 

Fig. 3.1. The probability of finding an electron with absolute momentum p in a hydrogen atom, observed by measuring the complete kinematics of ionisation events at the total energies shown (Lohmann and Weigold, 1981). The curve shows the square of the momentum-space wave function.

This article provides an introduction to the momentum perspective of the electronic structure of atoms and molecules. After an explanation of the genesis of momentum-space wave functions, relationships among one-electron position and momentum densities, density matrices, and form factors are traced. General properties of the momentum density are highlighted and contrasted with properties of the number (or charge) density. An outline is given of the experimental measurement of momentum densities and their computation. Several illustrative computations of momentum-space properties are summarized. 

An equally valid but different approach is to work with momentum-space wave functions n(yi,y2> 1 ) which depend upon the momentum-spin coordinates yj = (Pj, o ) jLi of the N electrons in the system but not on their positions. A momentum representation of the wave function does not yield any more or less information than the position representation of the wave function does. However, the momentum representation does provide a different perspective—one from the other end of Heisenberg s eyeglass. 

There are three distinct ways by which the momentum-space wave function can be obtained directly by solving either a differential or an integral equation in momentum or p space, or indirectly by transformation of the position-space wave function. 

In quantum chemistry, the state of a physical system is usually described by a wave function in the position space. However, it is also well known that a wave function in the momentum space can provide complementary information for electronic structure of atoms or molecules [1]. The momentum-space wave function is especially useful to analyse the experimental results of scattering problems, such as Compton profiles [2] and e,2e) measurements [3]. Recently it is also applied to study quantum similarity in atoms and molecules [4]. In the present work, we focus our attention on the inner-shell ionization processes of atoms by charged-particle impact and study how the electron momentum distribution affects on the inner-shell ionization cross sections. 

E. For the reciprocal-space solutions, ko represents the radius of the hypersphere onto which momentum-space is mapped by the generalized Fock transformation (equation (89)). As we saw above, the momentum-space wave functions are proportional to hyperspheri-cal harmonics on the surface of this hypersphere. The hyperspherical harmonics form a complete set, in the sense that any well-behaved function of the hyperangles can be expanded in terms of them. A set of hydrogenlike wave functions, all corresponding to the same value of ko (but with variable Z) is called a Sturmian basis [33-38,24] and such a basis set has the degree of completeness just mentioned. However, if Z is held constant while ko is variable within the set, then the continumn functions are required for completeness. 

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