Solving Scattering Problem: Get Help with Phase Shift Delta

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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?

 

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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.
 
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..
 

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"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.
 
exactly, it seems like is given in textbooks.. but getting through it is another thing
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
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