Momentum wave functions

Lowdin, P.-O., Angular momentum wave functions constructed by projection operators."... 

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 momentum density is given by the momentum wave functions and occupation number densities... 

One can construct wave functions by first coupling 1 and 2 and subsequently 3 or in any other order (Figure 5.1). The angular momentum wave functions corresponding to the two coupling schemes are related by a recoupling coefficient [Eq. (B.24)],... 

The use of real spherical harmonics is particularly bothersome. It has been demonstrated convincingly that the notion of geometrical sets of oriented real atomic angular momentum wave functions is forbidden by the exclusion principle. The use of such functions to condition atomic densities therefore cannot produce physically meaningful results. The question of increased density between atoms must be considered as undecided, at best. 

Just as angular momentum wave functions can be coupled together using the Clebsch Gordan coefficients, so too can spherical tensors. Two spherical tensors Rkl and S 2 can be combined to form a tensor of rank K which takes all possible values from (k + ki) to (k — k2), assuming k > kp. 

The momentum representation of the state vector of a system is its momentum wave function. 

In the following discussion, we shall review the momentum-space aspect of diatomic interactions, focusing upon the redistribution of the momentum density and its contribution to the interaction potential. Since all the readers may not be very familiar with the momentum-space treatment, we start with a brief introduction to the concepts of the momentum wave function and the momentum density. 

Alternatively, we can more easily obtain the momentum wave function ( p, ) from the usual position wave function P( r, ) by the 3N-dimensional Fourier transformation (Dirac, 1958),... 

Here we would like to add some comment of a general character. Dirac (1958) argued that the transformation from classical to quantum mechanics should be made, first by constructing the classical Hamiltonian in the Cartesian coordinate system and then by replacing the positions and momenta by their quantum-mechanical operator equivalents, which are determined by the particular representation chosen. The important point is that this transformation should be performed in the Cartesian coordinate system, for it is only in this system that the Heisenberg uncertainty principle for the positions and momenta is usually enunciated. In this connection, notice that some momentum wave functions such as those obtained by Podolsky and Pauling (1929) are correct wave functions that are useful in calculations of the expectation value of any observable, but at the same time they have a drawback in that the momentum variables used there are not conjugate to any relevant position variables (see also, Lombardi, 1980). 

Once the momentum wave function is given, the momentum density p(p) and the other reduced density matrices in momentum space are obtained by the same procedure as in position space (Lowdin, 1955 McWeeny, 1960 Davidson, 1976). For example, the momentum density is given by... 

Weigold, E., ed. (1982). Momentum Wave Functions 1982. American Institute of Physics, New York. 

On application of this equation it is found that the momentum wave functions for the harmonic oscillator have the same form (Hermite orthogonal functions) as the coordinate wave functions (Prob. 64-1), whereas those for the hydrogen atom afe quite different.1... 

Problem 64-1. Evaluate the momentum wave functions for the harmonic oscillator. Show that the average value of prx for the nth state given by the equation... 

Problem 64-2. Evaluate the momentum wave function for the normal hydrogen atom,... 

Abstract The momentum representation of the electron wave functions is obtained for the nonrelativistic hydrogenic, the Hartree-Fock-Roothaan, the relativistic hy-drogenic, and the relativistic Hartree-Fock-Roothaan models by means of Fourier transformation. All the momentum wave functions are expressed in terms of Gauss-type hypergeometric functions. The electron momentum distributions are calculated by the use of these expressions, and the relativistic effect is demonstrated. The results are applied for calculations of inner-shell ionization cross sections by charged-particle impact in the binary-encounter approximation. The reiativistic effect and the wave-function effect on the ionization cross sections are discussed. 

The momentum wave functions in various atomic models are calculated for arbitrary atomic orbitals. The nonrelativistic hydrogenic, the Hartree-Fock, the relativistic hydrogenic, and the Dirac-Fock models are considered. The momentum wave functions are obtained as a Fourier transform of the wave function in the position space. The Hartree-Fock and the Dirac-Fock wave functions in atoms are given in terms of Slater-type orbitals (STO s), i.e. the Hartree-Fock-Roothaan (HFR) method and the relativistic HFR (RHFR) method. All the wave functions in the momentum space can be expressed analytically in terms of hypergeometric functions. 

The momentum wave functions thus obtained are used to calculate inner-shell ionization cross sections by charged-particle impact in the binary-encounter approximation (BEA) [5]. The wave-function effect and the electronic relativistic effect on the inner-shell ionization processes are studied. 

The momentum wave function can be calculated analytically and expressed as... 

According to Rubinowicz [10], the relativistic momentum wave function is expressed as... 

The momentum distribution /(f) is expressed in terms of momentum wave function... 

From where having the inverse Fourier specialization of momentum wave function in terms of initial coordinate wave function (distribution)... 

The way we set up the Schrodinger equation makes it easiest to solve for the angular momentum wave functions in Table 3.1 first, but the Cartesian orbitals will be more useful when we get to molecules, where the orientation of the electron density along different Cartesian axes becomes important. 

In chemical equations for the reactimis in the atmosphere, atoms and molecules are sometimes denoted with symbols in a parenthesis, e.g. 0( P), CX D), O2 ( 2g ), O2 ( IIu), etc. These notation are called spectroscopic terms differentiating electronic states by the symmetry of angular momentum wave functions. Here, we do not enter into the theory of quantum chemistiy, the meaning and usefulness of the symbols and selection rules which is important for the discussion of photoabsorption and photolysis reactions are remarked. 

---