Fourier transforms of single-particle wave functions in cylindrical coordinates
Résumé
A formalism and the corresponding numerical procedures that calculate the Fourier transform of a single-particle wave function defined on a grid of cylindrical ( $\rho$ , z) coordinates is presented. Single-particle states in spherical and deformed nuclei have been chosen in view of future applications in the field of nuclear reactions. Bidimensional plots of the probability that the nucleon's momentum has a given value $K=\sqrt{k_{\rho}^{2}+k_{z}^{2}}$ are produced and from them the K -distributions are deduced. Three potentials have been investigated: a) a sharp surface spherical well (i.e., of constant depth), b) a spherical Woods-Saxon potential i.e., diffuse surface) and c) a deformed potential of Woods-Saxon type. In the first case the momenta are as well defined as allowed by the uncertainty principle. Depending on the state, their distributions have up to three separated peaks as a consequence of the up to three circular ridges of the bidimensional probabilities plots. In the second case the diffuseness allows very low momenta to be always populated thus creating tails towards the origin (K = 0). The peaks are still present but not well separated. In the third case the deformation transforms the above mentioned circular ridges into ellipses thus spreading the K-values along them. As a consequence the K-distributions have only one broad peak.