Time-Independent Schrodinger Solutions

Finding the energy levels and stationary wavefunctions of a quantum particle in a potential is the first task of computational quantum mechanics, solved either by shooting along the wavefunction or by diagonalizing a discretized Hamiltonian.

Definition

The time-independent Schrodinger equation is an eigenvalue equation whose solutions are the stationary states and energy levels of a quantum system; solving it numerically means finding those eigenvalues and eigenfunctions for a given potential.

Scope

This topic covers numerical solution of the stationary Schrodinger equation in one and a few dimensions: shooting and matching with eigenvalue search, the Numerov integration method, and matrix methods that discretize the Hamiltonian on a grid or in a basis. It treats bound states and, briefly, scattering states.

Core questions

• How does the shooting method find energy eigenvalues by enforcing boundary conditions?
• Why is the Numerov method well suited to integrating the Schrodinger equation?
• How does discretizing the Hamiltonian turn the problem into matrix diagonalization?
• How are the discrete bound states distinguished from the continuum?

Key theories

Shooting and matching

The wavefunction is integrated from the boundaries inward for a trial energy, and the energy is adjusted until the inward and outward solutions match smoothly, which selects the allowed eigenvalues.

Numerov integration

The Numerov method exploits the special structure of the Schrodinger equation, with no first-derivative term, to achieve high-order accuracy at low cost when integrating the wavefunction.

Matrix diagonalization of the Hamiltonian

Representing the Hamiltonian on a grid or in a finite basis yields a matrix whose eigenvalues are the energy levels and whose eigenvectors are the discretized wavefunctions, found by standard eigensolvers.

Clinical relevance

Solving the stationary Schrodinger equation gives atomic and molecular energy levels, the spectra of quantum wells and nanostructures, and the single-particle orbitals that feed electronic-structure calculations.

History

Numerical integration of the Schrodinger equation followed soon after its 1926 formulation, with the Numerov method, originally devised for celestial mechanics, becoming a staple; the growth of computers made full Hamiltonian diagonalization the routine alternative.

Key figures

• Boris Numerov
• Erwin Schrodinger
• Jos Thijssen

Seminal works

• thijssen2007
• giordano2006

Frequently asked questions

When should shooting be used instead of matrix diagonalization?

Shooting is natural and accurate for one-dimensional or radial problems where a single eigenvalue is sought at a time. Matrix diagonalization is more convenient when many levels are needed at once or in higher dimensions where shooting becomes awkward.

Why is the Numerov method preferred for this equation?

The Schrodinger equation has no first-derivative term, which the Numerov scheme is specifically designed to exploit, giving fourth-order accuracy with little extra work compared to a basic integrator.