Application of the Numerical Phase Method to Find the Electronic Energy Spectrum in a Spherical Quantum Dot with Arbitrary Confining Potential

Abstract

The paper discusses the phase method for solving a second-order differential equation, particularly the equation for eigenfunctions and eigenvalues of the Hamiltonian operator, i.e., the Schrödinger equation. The classical phase method is adapted to a problem containing two different regions with corresponding parameters. This method is implemented using computer algebra systems (Wolfram Mathematica). The modified method is applied to compute the electronic energy spectrum of an electron in a spherical quantum dot for cases of rectangular confining potential and smooth potential. For the rectangular potential, exact solutions of the corresponding Schrödinger equation exist, so they are compared with the results obtained from the proposed method, showing high accuracy in convergence. The advantage and value of the method lie in its applicability to spherical quantum dots, both single-layer and multi-layer, with arbitrary confining potentials.