### Quantum Harmonic Oscillator

#### Description

In exercise is designed to study the Quantum Harmonic Oscillator and to show the shooting method works. The bound states are of the form e-x2.

There are several important features to note:

1. The wave function is plotted with it's "zero line" rising along the vertical axis (to correspond to the shifting to the energy eigenvalue).

2. The walls of the graph are "soft" .. the potential changes as x changes.

3. It will always be possible to get a mathematical solution to the differential equation, but the important question for a physicist is "Does the solution have physical meaning?"  Solutions will have physical meaning if they satisfy the boundary conditions.

4. Left-click in the graph for graph coordinates.
Right-click in the graph to take a snapshot of the current graph.
Left-click-drag the mouse inside the energy level spectrum to change energy levels and wave function of the particle.

#### Question

Explain how you can observe that an energy value is acceptable as you right click-drag.

Answer: Right-click and drag in the energy spectrum and you should observe the right side of the wave function flipping from negative to positive.  This is a sign that you have passed through a physical solution to Schroedinger's equation, i.e., that the boundary condition at Y( x = +/- 1.00) =0. The correct boundary condition is assumed at the left side of the graph and the wavefunction is calculated in the direction of increasing x.

#### Question :

How do the spacing of the energy levels compare here to the infinite square well energy levels and the finite square well?

Related physlets : infinite square well and the finite square well

Credits : Physlet problems authored by Dan Boye. Modified by Scott Schneider.