Finite Well Simulation

Description:

This simulation calculates the wave functions and energies which are solutions to Schroedinger's equation for a particle inside a one dimensional finite square well (the sides of the well are not infinitely hard so that the wave functions can penetrate the walls).  


You can choose to find the solution for a chosen quantum number by entering a value of n at the lower right and clicking the 'Find' button. Holding the mouse down in the window shows the x location of the mouse and the energy. The green lines on the right show the energy levels you have calculated.

Don't change the potential yet!

Question 1:

Look at the wave functions for energy levels n = 1 through 5 by entering n at the lower right and clicking 'Find'. What can you say about the boundary conditions (what the wave function is doing when it reaches the edges of the potential) in this case as compared with the infinite square well? Note the energy level spacing (the green lines on the right- you can click on them to see which one is which); you will need them for Question 5.

Question 2:

Look at the wave functions for energy levels n = 8, 9, 10. At what energy level is the probability for finding the electron outside the well greater than finding it inside? What is the energy of this level? How does this energy compare with the energy depth of the well (click in the window at the top of the well to see the energy)?

Question 3:

Carefully note the amount of wave function penetration of the sides of the well for the n = 5 case. Now change both of the numbers 500 in the 'Potential' widow to 1500 and click 'Update' and then 'Find' the n = 5 wave function again. How does the amount of wavelength penetration change for the n = 5 case as the sides of the well get higher? What does this tell you about the penetration of the wave function for a well with infinite depth?

Question 4:

Now change both of the numbers 500 in the 'Potential' widow to 100 and click 'Update' and 'Find' the n = 5 wave function again. Compare this to the wave function for the case for a free electron where there is no potential well (change both 100s to 0s)? What can you say about the probability of finding the electron in each case? (Note: In the free electron case the electron wave function are usually moving- the wave functions calculated here are time independent or stationary state wave functions.)

Question 5:

Now erase everything in the 'Potential' window and replace that equation with 2000*x*x (the harmonic oscillator potential). Find the first 5 or so energy levels, noting the spacing between each level. What can you say about the spacing of the harmonic oscillator energy levels as compared with the square well? (Hitting the 'refresh' button on your browser will return the square well initial conditions.)