Timeline for Fourier Decomposition of Schrödinger's Equation with a Potential ${V}{\left({x}\right)}=e^x$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 30, 2021 at 3:26 | comment | added | Talmsmen | Thank you, I'll read into the theory and respond in a few days. Do you mind if I contact you by chat? | |
| Apr 30, 2021 at 2:58 | comment | added | mathstackuser12 | Not entirely sure what you mean here. However note that you have separated your PDE into two ODEs. the spatial ODE i think has solutions in terms of Bessel functions. Take those solutions and multiply them against the solutions for the temporal ODE to regain your time-dependence. Sturm-Liouville theory here is your friend by the way (assuming separation of variables is a valid approach to the problem at hand). | |
| Apr 30, 2021 at 2:15 | comment | added | Talmsmen | Would the appropriate formula be $\psi_{t}=i\left(\psi_{zz}z^2 + \psi_{z}z - z \psi\right)$? Then, wave functions that solve Bessel's differential equation would serve as steady-state solutions correct? Is there any corresponding PDE that would allow me to solve for arbitrary values of time. This problem is closely related to the Toda Lattice and inverse scattering transform. | |
| Apr 30, 2021 at 1:51 | comment | added | mathstackuser12 | if you make a change of variables in your spatial equation $z={{e}^{x}}$ then you get an something that looks a lot like a Bessel function. Pretty sure you can then get these to pop out of it which if so, there's your classical orthogonal set and you're possibly looking at Hankel transforms (?) (polar version of Fourier). | |
| Apr 29, 2021 at 21:31 | review | First posts | |||
| Apr 29, 2021 at 21:47 | |||||
| Apr 29, 2021 at 21:31 | history | edited | Talmsmen | CC BY-SA 4.0 |
Corrected Typos
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| Apr 29, 2021 at 21:30 | history | undeleted | Talmsmen | ||
| Apr 29, 2021 at 21:00 | history | deleted | Talmsmen | via Vote | |
| Apr 29, 2021 at 20:58 | history | asked | Talmsmen | CC BY-SA 4.0 |