Bound State Solutions to the Schroedinger Equation

The following applet lets you explore the dependence of the energy levels and wave functions on the shape, depth, and width of a potential well:

http://www.fen.bilkent.edu.tr/~yalabik/applets/1d.html

With the mouse you can sketch a 1-D potential well and the applet will plot the corresponding energy level diagram and wave functions for the first 4 levels. Absolute numbers are not given and there are maximum and minimum values of potential that can be drawn. Explore the following potentials.

Square well:  Note that the wave function extends beyond the walls of the well.  Note that the spacing of the levels increases with increasing quantum state, as is the case for an infinitely deep well.  Note also how the energy levels and wave functions change as you change the width of the well (for a fixed depth).  As you narrow the well the energy levels increase, as suggested by the formula for the energy levels for an infinite well.  If the width is too small, then the highest states are free (not quantized).  Note the effect of changing the depth of the well.

Harmonic oscillator:  Sketch a quadratic well.  (The walls will have a finite height.)  Note that the spacing of the levels is approximately equal, as predicted for the harmonic oscillator.

Hydrogen atom:  Sketch a potential approximately like the Coulomb potential (V ~ 1/r).  (You won't be able to make it infinitely deep.)  Note that the spacing of the energy levels decreases with increasing quantum number, as predicted for the hydrogen atom.  The wave functions will not be like the actual radial wave functions for the H-atom since this is only a 1-D potential.