Two charges on plane in a magnetic field: III. $He^+$ ion

TL;DR

This paper calculates the ground state energy of a $He^+$ ion on a plane under a magnetic field, considering finite nuclear mass, and finds a sharp energy change at a critical pseudomomentum.

Contribution

It introduces a variational method to compute the $He^+$ ion's ground state energy across a wide range of magnetic fields and pseudomomenta, accounting for finite nuclear mass effects.

Findings

Ground state energy computed for magnetic fields from 0.01 to 100 a.u.

Identified a sharp energy change at a critical pseudomomentum $K_c$.

Achieved high accuracy, matching results with the Lagrange-mesh method.

Abstract

The $He^+$ ion on a plane subject to a constant magnetic field $B$ perpendicular to the plane is considered taking into account the finite nuclear mass. Factorization of eigenfunctions permits to reduce the four-dimensional problem to three-dimensional one. The ground state energy of the composite system is calculated in a wide range of magnetic fields from $B=0.01$ up to $B=100$ a.u. and center-of-mass Pseudomomentum $K$ from $0$ to $1000$ a.u. using a variational approach. The accuracy of calculations for $B = 0.1 $ a.u. is cross-checked in Lagrange-mesh method and not less than five significant figures are reproduced in energy. Similarly to the case of moving neutral system on the plane a phenomenon of a sharp change of energy behavior as a function of $K$ for a certain critical $K_c$ but a fixed magnetic field occurs.