[Paper Review] Numerov numerical method applied to the Schrödinger equation

Francisco Caruso, V. Oguri|arXiv (Cornell University)|Mar 17, 2014

Quantum Mechanics and Non-Hermitian Physics6 references3 citations

TL;DR

This paper presents a didactic application of the Numerov method to solve eigenvalue problems in quantum mechanics, specifically for the 1D harmonic oscillator and hydrogen atom. It extends the standard Numerov method to handle first-derivative terms in the Schrödinger equation, validates numerical results against analytical solutions, and demonstrates high accuracy in computing energy levels and radial wavefunctions using a second-order finite difference scheme.

ABSTRACT

In this paper it is shown how to solve numerically eigenvalue problems associated to second order linear ordinary differential equations, containing also terms which depend on the variable. A didactic presentation of the Numerov Method is given and, in the sequel, it is applied to two quantum non-relativistic problems with well known analytical solutions: the simple harmonic oscillator and the hydrogen atom. The numerical results are compared to those obtained analytically.

Motivation & Objective

• To provide a didactic introduction to the Numerov method for solving second-order linear ODEs in quantum mechanics.
• To extend the standard Numerov method to handle terms involving the first derivative of the wavefunction.
• To numerically solve the time-independent Schrödinger equation for two benchmark quantum systems: the harmonic oscillator and hydrogen atom.
• To compare numerical eigenvalues and wavefunctions with analytical solutions to validate the method's accuracy.
• To demonstrate the method's robustness and general applicability in modern quantum mechanical calculations.

Proposed method

• The Numerov method is applied as a second-order finite difference scheme to discretize the Schrödinger equation, avoiding explicit use of first derivatives.
• A modified Numerov formula is derived to include terms dependent on the first derivative of the wavefunction, enabling application to the hydrogen atom with angular momentum.
• The method uses a three-point stencil to propagate the wavefunction forward and backward from initial conditions, ensuring stability and accuracy.
• Boundary conditions are applied at both ends of the spatial domain (e.g., origin and infinity) to solve the eigenvalue problem.
• The algorithm is implemented in C++ using the ROOT framework, with code available via a public link.
• The method is validated by comparing numerical energy levels and radial wavefunctions with exact analytical results.

Experimental results

Research questions

• RQ1 How accurately can the Numerov method compute the energy eigenvalues of the 1D harmonic oscillator when extended to include first-derivative terms?
• RQ2 Can the modified Numerov method correctly reproduce the radial wavefunctions and energy levels of the hydrogen atom in 3D?
• RQ3 What is the numerical error in the computed eigenvalues when compared to analytical solutions for both the harmonic oscillator and hydrogen atom?
• RQ4 How does the method perform in solving singular boundary value problems involving the Schrödinger equation with non-zero angular momentum?
• RQ5 To what extent can the Numerov method be generalized and applied to modern quantum mechanical problems beyond standard textbook cases?

Key findings

• The numerical energy levels for the 1D harmonic oscillator and hydrogen atom (l=1) agree closely with analytical solutions, with deviations less than 2%.
• For the hydrogen atom (l=1), the computed energy levels were E2 = -3.47 eV (analytical: -3.40 eV), E3 = -1.54 eV (analytical: -1.51 eV), and E4 = -0.83 eV (analytical: -0.85 eV).
• The radial wavefunctions obtained numerically matched the analytical solutions in shape and node structure, confirming the correctness of the method's implementation.
• The modified Numerov method successfully handled the first-derivative term in the Schrödinger equation for the hydrogen atom by incorporating the effective potential and radial derivatives.
• The method demonstrated second-order convergence and stability over large spatial domains, including singular behavior at the origin.
• The implementation in C++ using ROOT proved efficient and suitable for educational and research applications in quantum mechanics.