Solving Scattering Problem: Get Help with Phase Shift Delta

[ftft]

hi all,

I am facing a confusing problem with scattering theory and any hint would be greatly appreciated..
In the famous scattering by a square well problem the phase shift$$\delta$$ is a smooth curve in the particle energy... but what I get from my own derivation is not exactly as it should be (see attached file)...

any hint?

 

[Meir Achuz]

Your phase shift has jumped by pi, probably at the energy where a bound state appears.
The bound state pulls the wave function in, changing the phase by pi.

 

[ftft]

Thank you.. however the figure doesn't show the expected behaviour at small energies...
I attached a pdf file showing what I did exactly, hoping that somebody will discover what the problem is in my derivation..
I am really desperate to know..

 

[Meir Achuz]

"I couldn't get the correct behavior of the curve at small energies E at any
value of the potential.
I think my derivation is correct and also my variable…. So why doesn't it
give me the supposed figures? Where is the error?"

I don't know what you mean by the above statements.
Your figure in the thumbnail looks like what I would expect.
I can't (won't) go through your steps in detail, but they seem like what I have seen in textbooks.

 

[ftft]

exactly, it seems like is given in textbooks.. but getting through it is another thing