## Section 7.4: Exploring Energy Eigenfunctions Using the Shooting Method

A particle is confined to a box with hard walls at x = −3 and x = 3 and an unknown potential energy function within the box. Change the energy slider and examine the solutions to the time-independent Schrödinger equation for this system.  In the animation, ħ = 2m = 1.  Restart.

1. Determine the energy of the ground state. Start by entering the energy value 4.86 in the slider text box. Use a procedure similar to the following: Enter 4.86. Enter 4.861, 4.862, .... What does the energy eigenfunction look like when you over/under shoot the energy?
2. How many energy eigenfunctions (states that also satisfy the boundary conditions) are between E = 0 and E = 20?
3. Determine the energy eigenvalues for the system between E = 0 and E = 20.
4. Examine each eigenfunction and sketch a reasonable guess for the potential energy function. Make sure that you also explain your reasoning for the functional form of your sketch.

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