Single particle in a reflection-asymmetric potential

TL;DR

This paper investigates single particles in reflection-asymmetric potentials by solving the Schrödinger equation using two numerical methods, demonstrating their efficiency and consistency in calculating single-particle energies with high precision.

Contribution

It introduces and compares two numerical techniques for solving the Schrödinger equation in reflection-asymmetric potentials, addressing divergence issues and demonstrating high accuracy.

Findings

Numerical techniques effectively solve the Schrödinger equation in complex potentials.

Single-particle energy levels are accurately computed with differences less than 10^{-4} ħω₀.

The methods are validated by consistent results in spherical and reflection-asymmetric cases.

Abstract

Single particles moving in a reflection-asymmetric potential are investigated by solving the Schr\"{o}dinger equation of the reflection-asymmetric Nilsson Hamiltonian with the imaginary time method in 3D lattice space and the harmonic oscillator basis expansion method. In the 3D lattice calculation, the $\bm{l}^2$ divergence problem is avoided by introducing a damping function, and the$\langle \bm{l^2}\rangle_N$ term in the non-spherical case is calculated by introducing an equivalent $N$-independent operator. The efficiency of these numerical techniques is demonstrated by solving the spherical Nilsson Hamiltonian in 3D lattice space. The evolution of the single-particle levels in a reflection-asymmetric potential is obtained and discussed by the above two numerical methods, and their consistency is shown in the obtained single-particle energies with the differences smaller than 10$^{-4}~[\hbar\omega_0]$.