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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 ([tex]A_{I}, B_{I}, A_{II}, B_{II}[/tex] 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 ([tex]A_{IV}[/tex] over the incident coefficient ([tex]A_{I}[/tex]), right?
Any suggestions, critiques are welcome. Thanks all!
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 ([tex]A_{I}, B_{I}, A_{II}, B_{II}[/tex] 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 ([tex]A_{IV}[/tex] over the incident coefficient ([tex]A_{I}[/tex]), right?
Any suggestions, critiques are welcome. Thanks all!
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