Timeline for Localization of solutions for time-dependent Schroedinger equation
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 7, 2019 at 16:17 | answer | added | Christian Remling | timeline score: 1 | |
| Mar 6, 2019 at 19:46 | comment | added | Christian Remling | Or you could fix a positive time (obtained from the distance of the supports presumably) and then ask about comparison of the two solutions at that time. | |
| Mar 6, 2019 at 19:45 | comment | added | Christian Remling | Q2 sounds quite interesting potentially, but I think we need to be careful with the quantifiers here. Since $e^{itH}\to 1$ strongly as $t\to 0$, it is of course trivially true that the solutions to any two equations will be close at small times (since both will be close to the initial value). It does make sense though to ask about bounds that hold uniformly across all potentials satisfying the support condition. | |
| Mar 6, 2019 at 15:22 | comment | added | Ivan | @ChristianRemling I think that the RAGE theorem is not what I'm looking for exactly, since I am interested to see that the solution "does not interact" with the potential under some conditions, which seems more reasonable for small times. But I will have a closer look, thank you. And yes $\langle \psi,x \psi\rangle$ is perfectly reasonable =) I just don't think that it captures this idea of locality | |
| Mar 6, 2019 at 15:10 | comment | added | Ivan | I edited slightly the question. Basically I would like to see that as long as the particle is far away from the barrier, it does not "see it", which means that there is some property of some particular solution that holds independently of the potential, but maybe just for small times. @KonstantinosKanakoglou L^2-norm is perfectly fine, any observable would do fine as long as it remains finite for small times. I was trying to prove this result for some weighted norms or for averages of some simple compactly supported observables, but I was not able too. | |
| Mar 6, 2019 at 14:28 | history | edited | Ivan | CC BY-SA 4.0 |
added 100 characters in body
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| Mar 5, 2019 at 21:28 | comment | added | Konstantinos Kanakoglou | I am also a little confused by what you mean "in some norm". Aren't we speaking about $L^2$ - wave functions? | |
| Mar 5, 2019 at 21:27 | comment | added | Konstantinos Kanakoglou | About your second question: are you actually asking whether the solution for an arbitrary potential $V(x)$ is always "close" to the free particle solution? If this is what you are asking, i think the answer is evidently no. | |
| Mar 5, 2019 at 17:59 | comment | added | Christian Remling | It is perfectly reasonable to study something like $\langle \psi, x\psi\rangle$ also (assuming it exists), and there is a large literature on this too (key word "quantum dynamics" perhaps). | |
| Mar 5, 2019 at 17:58 | comment | added | Christian Remling | Whether or not the $L^2$ norm spreads out at large times is closely related to the spectral properties of $-\Delta+V$ (sometimes grandiosely called the RAGE thm, but it's really just basic facts about the Fourier transform in disguise). | |
| Mar 5, 2019 at 17:43 | history | asked | Ivan | CC BY-SA 4.0 |