Wave functions space

When higher n Feynman paths contribute to the wave function, one has simply to apply repeatedly the single- to double-space mapping, until the nuclear wave function is completely unwound (in the sense just defined). Thus, if the wave function contains only n = —2, —1,0,1 paths, then we need to compute a function in the double space that satisfies the boundary condition 4 (([)) = —(cj) + 4ti). Adding this function to the [which satisfies he(( )) = he(( ) + 4ti)] then gives a new function, [ 4(( )), which occupies the quadruple space (j) = 0 8ti (see Fig. 18c). This new quadruple-space wave function will be completely unwound, such that there is a gap between its clockwise and counterclockwise branches. The n= 2, 1,0,1 contributions will lie in the... 

In a numerical calculation, the number of times that one can unwind the will be limited by the maximum size of cover space that can be treated computationally. An efficient way to unwind onto an 2hn cover space will be to compute the h single-space wave functions that satisfy the boundary conditions... 

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

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... 

Both W(x, t) and A p, i) contain the same information about the system, making it possible to find p) using the coordinate-space wave function W(x, t) in place of A(p, i). The result of establishing such a procedure will prove useful when determining expectation values for functions of both position and momentum. We begin by taking the complex conjugate oi A p, i) in equation (2.8)... 

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. 

I. Real Space Wave Functions of f Electrons in Condensed Matter. 

A discussion of the real space wave functions for RE and actinides, showing the large differences with the atomic like wave functions can be found, for example, in Freeman s introductory paper to the Physics of Actinides and Related 4f Materials (11), (especially Figure 8 of that paper for y - U). See also (12). In our analysis here we have concentrated on the rare earths. 

The real space wave functions are the solutions of a Schrodinger like equation... 

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.

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