Solving arbitrary potential barrier problems

[IHateMayonnaise]

I am doing a computational project in my undergraduate Quantum Physics course on tunneling through a potential barrier. But, it's an irregular potential barrier, so I cannot simply use the results from a textbook. The first diagram, with corresponding wave equations, are shown in the first attached image (Sorry in advance for my messy handwriting). Now, I am pretty sure everything I have there is right, however I wouldn't be surprised if I made some mistakes as I only went through it once. So, if you see any mistakes, please let me know!

As I said, I am doing the project on tunneling through this barrier. So, I need to find an expression for the inverse of the transmission coefficient (1/T), and to do this I need to use boundary conditions and then solve for the coefficients (A_{I}, B_{I}, A_{II}, B_{II} etc...) in terms of ratios of one another. I have the equations from the boundary conditions (on the second attached image), so, all I need to do is do lots of algebra and then I should have my expression.

At first I tried to reduce these six equations using linear algebra, since I assumed this would be easier. However, it wasn't too long before this too became too cumbersome. My first question is: is there a program out there that can solve these equations automatically? I was not able to find one that worked for such complicated expressions (even when I substituted for simpler values).

Furthermore, I have tried to do the derivation for 1/T on a standard potential barrier, however each time I made too many mistakes to make it. Here is my second question: If I were to (somehow) evaluate my coefficient ratios, to find 1/T I would simply square the ratio of the transmitted coefficient (A_{IV} over the incident coefficient (A_{I}), right?

Any suggestions, critiques are welcome. Thanks all!

 

[eys_physics]

I can not see the images. Can you try to upload them again?

 

[ZapperZ]

The attachments have been approved.

I think you may not be approaching this correctly. You probably should solve the differential equation directly by numerical methods, rather than solving the wavefunction. So rather than getting a function, all you'll get is a series of values for it.

I also think this is a rather difficult profile to start with. Did anyone come up with this particular profile for the potential? You at least want something that has a physical significance.

Zz.

 

[IHateMayonnaise]

I think you may not be approaching this correctly. You probably should solve the differential equation directly by numerical methods, rather than solving the wavefunction. So rather than getting a function, all you'll get is a series of values for it.

I also think this is a rather difficult profile to start with. Did anyone come up with this particular profile for the potential? You at least want something that has a physical significance.

Zz.

Can't I just solve them explicitly, and find explicit values (since they depend only on constants)? Or do you think it would just be easier to approximate them numerically using Runga-Kutta?

My professor suggested this barrier..I can imagine it's probably pretty difficult to find a simple example like this that has physical significance, since it is only in two dimensions. However, if I can pound this out fairly quickly I could most likely try a more complicated example; for instance a particle in a three dimensional box.

 

[ZapperZ]

This may be a bit too late, but you might want to check the latest issue of American Journal of Physics.

AJP v.76 (2008).

It is an issue devoted to computational physics, and contains several examples of various computational projects at the undergraduate level.

Zz.