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In many books, we find the proof of the time-independent Schrödinger equation which can be derived from the wave equation and the expression of de Broglie wavelength. But what about the time-dependent equation? In ion-atom collisions, there is a way to prove the eikonal equation which is similar to the TDSE where the starting point is TISE, if we consider that the projectile ion does linear trajectories, time can be embedded and we prove the eikonal equation that very known in a semiclassical treatment of ion-atom collisions. Is there a similar proof for example, and how time is introduced?

Qmechanic
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Malek_Physics
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Good question. I don't know about the eikonal equation but there are many experiments which essentially test the momentum-space wave function. Look up (e,2e) momentum spectroscopy, and recoil ion momentum spectroscopy. However, as a whole, position space checks of the wave function are pretty rare, and I'm not aware of a single full imaging of a wave function which measures the phase and amplitude, except in simplified systems. Maybe someone knows better than I do in this regard. Therefore I don't have a complete answer to your question but maybe these experiments can lead you on the right track.

doublefelix
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  • Thank you for you response. I meant by my question a mathematical proof. The eikonal eqution in optics gives the limit of angular interpretation and it is obtained for small wavelengths. Analogously, when the projectile has a high momentum the corresponding wavelength is then very short. Therefore the projectile may not scatter and follow rectilinear paths ($z=b+v*t$). You can check this paper link. – Malek_Physics Apr 16 '20 at 14:56
  • "Look up (e,2e) momentum spectroscopy...". The most theoretical papers on the assisted collision (e,2e) used a perturbative FBA and/or SBA. Where the results don't show a good agreement with the experiments and a non-perturbative treatment is hard to implement especially for assisted collisions. – Malek_Physics Apr 16 '20 at 15:12
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    I see, if you wanted a mathematical proof then the link by ZeroTheHero is my best understanding of the answer to that as well. I will check out the paper you linked – doublefelix Apr 16 '20 at 23:14
  • Thank you, you're right. – Malek_Physics Apr 16 '20 at 23:35