# Solving arbitrary potential barrier problems

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!

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

ZapperZ
Staff Emeritus
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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.

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
Staff Emeritus