# Quantum Single Well Energy Levels Java Application

## Physics Background

Solutions of the time-independent Schrödinger equation
determine the wave function of an electron bound in a potential well.
The potential function V(x) is assumed to exhibit a central well
region where the electron energy, *E*, is higher than the floor of the well,
but also climbs to a field potential that is higher then the
electron energy, so that the electron is bound inside the well.
It is a consequence of the wave solutions that there are only certain
energy levels that allow the wave solutions to resonate within the structure
of the potential well. In order for it to correspond to
a real physical particle, the wave function solution must remain bounded
at the extremes of the *x* coordinate so that the wave is
normalizable over all *x*.

There are two plot modes available, one showing the energy levels,
and the other plotting the wave function.

In the wave function mode, the numerical solution of
the Schrödinger equation starts at the left hand edge and progresses
to the right, as the *x* value increases. By adjusting the
electron energy you can observe that there are only a few values
of *E* that initiate a solution that will end up giving a
normalizable wave function at the right hand edge. These energy levels
of course correspond with the values with the allowed levels shown
in the energy level plot mode. Those allowed levels in blue are
solved numerically by the application directly from the Schrödinger equation
with a search algorithm that converges
automatically on the *E* values that end up satisfying a condition
that indicates the solved wave function is normalizable.

## The Quantum Single Well Energy Levels Application

Click to Download

Run from a shell or command terminal in the download directory with:

java -jar EnergyLevelsSingle.jar

This application lets you find allowed energy levels for an electron in a potential well.
Various profiles of the well potential can be chosen: Rectangular, Triangular, Coulomb
and Quadratic. The quadratic profile corresponds to the quantum harmonic oscillator.

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