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--- cwf:space-analyticity_of_wave_function:free_evolution [2016/11/23 16:10] – markus
+++ cwf:space-analyticity_of_wave_function:free_evolution [2016/11/25 15:19] (current) – markus
@@ Line 18: / Line 18: @@
 **Corollary.** If for some $a>0$ the initial state $\Psi_0$ fulfils all the properties of $\hat f$ in the theorem above then the same properties hold for $\Psi$ as the solution of the free Schrödinger equation \eqref{eq-schroedinger} at all times.
-**Proof.** We clearly set $\hat f = \Psi_0$ in the Paley-Wiener theorem above, then the inverse Fourier transform of $\Psi_0$ and thus also the direct Fourier transform $\hat \Psi_0$ (which is identical to the inverse one modulo a parity switch $k \mapsto -k$) has $\mathrm{e}^{b|k|}\hat\Psi_0 \in L^2(\mathbb{R}^n)$ for all $b<a$. Now the free evolution for a time period $t$ in Fourier space is just a multiplication with the phase factor $\mathrm{e}^{-\mathrm{i}tk^2/2}$ which will not change this $L^2$ condition. This means when taking the Fourier transform to go back to ordinary space the Paley-Wiener theorem becomes applicable once more and $\Psi(t)$ has just the same properties as the inital state.
+**Proof.** We clearly set $\hat f = \Psi_0$ in the Paley-Wiener theorem above, then the inverse Fourier transform of $\Psi_0$ and thus also the direct Fourier transform $\hat \Psi_0$ (which is identical to the inverse one modulo a parity switch $k \mapsto -k$) has $\mathrm{e}^{b|k|}\hat\Psi_0 \in L^2(\mathbb{R}^n)$ for all $b<a$. Now the free evolution for a time period $t$ in Fourier space is just a multiplication with the phase factor $\mathrm{e}^{-\mathrm{i}tk^2/2}$ which will not change this $L^2$ condition. This means when taking the Fourier transform to go back to ordinary space the Paley-Wiener theorem becomes applicable once more and $\Psi(t)$ has just the same properties as the inital state. $\Box$
+If the initial state is assumed analytic but the evolution is not free, including a non-analytic or even singular potential (like the Coulomb central potential), one would suppose that the analytic property gets destroyed. This case will be discussed on a [[coulomb_potential|separate page]].