# Energy Eigenfunctions and the Shooting Method

The Shooting Method model solves the time independent Schrödinger equation with potential energy V(x) when the energy E is changed.

Not all solutions this differential equation are valid. We seek solutions y(x) that are zero outside of a finite spatial region [xmin, xmax] and we require that the solution be zero at the boundaries in order to avoid a discontinuous jump in y(x). A solution to the time independent Schrödinger equation that satisfies these conditions is known as an energy eigenfunction.

Adjust the energy E to find the energy eigenfunctions for various potential energies. The differential equation is solved starting at the left boundary when the slider is moved. (Plank's constant h and mass m are one.) The error in y(x) at the right boundary is shown as a red line. The computation is automatically terminated if the solution becomes larger than 109 because a valid eigenfunction must be finite and normalizable.

## Reference:

Gould, Tobochnik, and Christian An Introduction to Computer Simulation Methods 3ed page 677.

### Note:

This simulation was created using the Easy Java Simulations (Ejs) modeling tool. You can modify this simulation if you have Ejs installed by right-clicking within a plot and selecting "Open Ejs Model" from the pop-up menu item. Information about Ejs is available at: < http://www.um.es/fem/Ejs/ >