The first code snippet allows you to simulate the wave function starting the finite difference method either from the left or from the right boundary. If the numeric method is working, we do not expect any difference between the two solutions. The error should be so small that it is not noticeable on a plot. However, the number of points and the distance between the points does matter. To check the influence of the number of points, perform the following simulations and take a screengrab or export each plot for your report and answer the questions associated with the simulations:

 1.) Start the simulation with the default number of points (n = 10). Do the two solutions match on the plot?  

2.) Reduce the number of points to n < 10. Do the solutions still match?  

3.) Increase the number of points to n > 10. Do you notice any difference in matching from the simulation in (1)?  

4.) Perform a simulation for n = 50. Why is the plot being cut-off vertically? Should this be the case? (Remember, that we are plotting the wave function and the wave function square has a physical meaning.) 

 

Task 2:

You noticed that the original code has issues. Without giving away the answer to Task 1, Question 4, this issue can be fixed easily and this has been done in the last code segment on the website or in the Python script "Numerov_script2.py". In this task,

 

1.) Re-run the simulation for a number of points (n > 10) that you consider sufficient to represent the waveform with no visible deviation between the finite difference method starting at the left and the one starting at the right boundary. Take a screengrab or export the plot for your report.

 

There is another issue with the code provided on the website. It only simulates the wave function for the ground state of the infinite well. However, you can change the code to simulate the wave function for the first excited state.

 

2.) Change the code in the Python script "Numerov_script2.py" to simulate the first excited state (Quantum number n = 2) of the infinite quantum well. Run the simulation for a number of points (n > 10) that you consider sufficient to represent the waveform with no visible deviation between the finite difference method starting at the left and the one starting at the right boundary. Take a screengrab or export the plot for your report. 

3.) Bonus: Repeat (2) for the second excited state (Quantum number n = 3) of the infinite quantum well. 

 

Task 3:

To check what happens if we make the walls of the well steeper, approaching the infinite well, you should perform the following simulations and take a screengrab or export each plot for your report and answer the question associated with the simulations: 

 

1.) Perform simulation for at least the first three energy levels for the potentials of V = x2 , V = x4 and V = x6 . What happens to the spacing of the energy levels with increasing power?

Answer rating: 100% (QA)

Task 1 1 Start the simulation with the default number of points n 10 Do the two solutions match on the plot You should run the simulation using the provided code with n 10 The two solutions should mat... View the full answer