NDodbele Final Project Submission

[NDodbele]

Here is the notebook for my final project!

[gforsyth]

Hey @NDodbele , really nice commit history there. You also win a prize for good commit messages.

You can remove that Test file you created with a

git rm Test.txt
git commit

[ncclementi]

Nikita, your project is really interesting. However, the matrix set up, is not correct. That's the reason why you are getting weird numerical solutions. They should look like the analytical ones.

The discretization is right, but you are not writing properly the matrix form. I'm attaching something that should help you to fix the problems you are having.

[labarba]

Nice clever title, Nikita, but I do contend that an introductory explanation of Schroedinger's paradox should not come from a Hollywood screen writer. It's nice to have a humorous hook to a serious work, but perhaps it was unnecessary here.

Your discretization, shown in equation (5), is correct. However, from there to the matrix form in equation (7), something definitely went awry.

Like Naty said in her comment, an easier way to go about the job here would have been to re-arrange your Schroedinger equation so it looks just like a diffusion equation. Then you can follow the workings, step by step, in Module 4 of the course.

Instead, you decided to "set" the symbol sigma to a combination of parameters appearing inside the parenthesis in your discretized equation (6), but not including the imaginary number i. Then in your matrix construction of equation (7), you have terms with this symbol sigma, but no imaginary number: your matrix is not complex-valued, although your right-hand-side is. I cannot figure out where you got that matrix form—perhaps you tried to emulate a matrix form in one of your references, possibly the thesis of Sehra (2006), where the matrix H in page 18 looks somewhat similar to yours; but they are using something called Cayley's form and later multiply that matrix H by the imaginary number and subtract from the identity.

In sum, your matrix system is incorrect. You should've just trusted yourself that you could do it by yourself, following our example in Module 4!

Why did you pick that initial condition? What does it mean?

Next, you simply assign a number to your symbol sigma—it seems you have mistakenly confused this combination of parameters with a CFL condition, simply by choosing to use sigma as your symbol. There is no reason to set that combination of parameters to 0.5 here: it is not a CFL number!

And when you get a solution, in code cell In[8], you print it out, and are happy to just see complex numbers come out ... how do you know that is the correct solution? (in fact, it is incorrect).

The output that you see plotted in Out[9] and Out[10] is an answer, but it is not the right answer. It can't be ... you coefficient matrix is incorrect. But you argued pretty confidently that you're seeing a wave packet—where did you get that from?

When you plot your probability density functions, in Out[13] and Out[20], there is an issue with the title running into the scaling factor of the vertical axis, so we can't read it. But it's clear that they are not the same factors ...

By the way, if you add a semi-colon at the end of a plotting code-block, you get rid of that ugly <matplotlib ... > output.

Typos, grammar, etc.

a parabolic partial differential equation which is the basis for —> that is the basis
We will consider an electron which is traveling inside a region —> that is traveling
See:
http://www.getitwriteonline.com/archive/103103whichthat.htm
http://www.writersdigest.com/online-editor/which-vs-that

the potential at the walls are infinite —> is infinite
whats going with our wave packet —> what's going on
fameous —> famous
Hamiltoian —> Hamiltonian