Exact Neutron-Proton Wavefunctions Using the Phase Function Method

TL;DR

This work addresses the problem of obtaining exact two-body scattering wavefunctions directly from phase-shift data. It applies the Phase Function Method (PFM) with inverse Morse potentials, whose parameters are optimized against the GRANADA np partial-wave analysis, to compute distance-dependent quantities $\delta(r)$, $A(r)$, and exact wavefunctions $u(r)$ for uncoupled neutron–proton channels up to $r=5$ fm across seven laboratory energies $E_{\text{lab}}$. The results show excellent agreement with Nijmegen-II, validating the approach and illustrating its transparency in elucidating short- and mid-range interaction effects. The study provides a robust, extensible framework for generating realistic scattering wavefunctions directly from inverse potentials and sets the stage for incorporating coupled-channel dynamics in future work.

Abstract

Radial phase shifts ($δ(r)$), amplitude functions ($A(r)$), and exact wavefunctions ($u(r)$) for various uncoupled S, P, and D channels of neutron--proton scattering have been calculated using the Phase Function Method (PFM). In these calculations, inverse potentials obtained from the Morse function as the zeroth-order reference potential are employed. The parameters of the Morse potential were optimized using the comprehensive GRANADA partial wave analysis, consisting of 6713 experimental \textit{np} phase shift data points from 1950 to 2013, by minimizing the mean square error (MSE) as a cost function. The present work provides detailed radial dependence of $δ(r)$, $A(r)$, and $u(r)$ up to 5~fm for laboratory energies $E_{\ell \text{lab}} = [1, 10, 50, 100, 150, 250, 350]$~MeV. The obtained wavefunctions show excellent agreement with high-precision Nijmegen-II results, highlighting the accuracy and transparency of the PFM approach for uncoupled scattering states.

Figures (8)

• Figure 1: Detailed flowchart for obtaining phase shift, amplitude function and wavefunction.
• Figure 2: (top)Phase shifts $\delta(E)$ for various uncoupled scattering states are shown with respect to energy $E(MeV)$. (bottom) Respective interaction potentials for various scattering states.
• Figure 3: Phase function, amplitude function, and radial wavefunctions for the $^{1}S_{0}$ channel. Panel (a) shows the variation of the phase shift $\delta_{0}(r)$, panel (b) the amplitude function $A_{0}(r)$, and panels (c--f) the corresponding wavefunctions $u_{0}(r)$ at $E_{\text{lab}} = 10,\; 50,\; 150,$ and $250$ MeV. Results obtained using PFM are compared with Nijmegen-II calculations.
• Figure 4: Phase function, amplitude function, and radial wavefunctions for the $^{1}P_{1}$ channel. Panel (a) shows the variation of the phase shift $\delta_{0}(r)$, panel (b) the amplitude function $A_{0}(r)$, and panels (c--f) the corresponding wavefunctions $u_{0}(r)$ at $E_{\text{lab}} = 10,\; 50,\; 150,$ and $250$ MeV. Results obtained using PFM are compared with Nijmegen-II calculations.
• Figure 5: Phase function, amplitude function, and radial wavefunctions for the $^{3}P_{0}$ channel. Panel (a) shows the variation of the phase shift $\delta_{0}(r)$, panel (b) the amplitude function $A_{0}(r)$, and panels (c--f) the corresponding wavefunctions $u_{0}(r)$ at $E_{\text{lab}} = 10,\; 50,\; 150,$ and $250$ MeV. Results obtained using PFM are compared with Nijmegen-II calculations.