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

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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.

- 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?
- How many energy eigenfunctions (states that also satisfy the boundary conditions) are between
*E*= 0 and*E*= 20? - Determine the energy eigenvalues for the system between
*E*= 0 and*E*= 20. - 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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