4.4 Exemples

1. $\psi (x)=\left\{ \begin{array}{ll} 1 & \left\vert x\right\vert <\frac{1}{2\alpha }\\ 0 & \mathrm{ailleurs} \end{array}\right.$ $\longrightarrow \; \tilde{\psi }(k)=\frac{1}{\sqrt{2\pi }}\frac{\sin \frac{k}{2\alpha }}{k/2}$
2. $\psi (x)=e^{-\alpha \left\vert x\right\vert }\; \... ...frac{1}{\sqrt{2\pi }}\frac{1}{\alpha }\frac{2}{1+\left( k/\alpha \right) ^{2}}$ (Lorentzienne)
3. $\psi (x)=e^{-\alpha ^{2}\frac{x^{2}}{2}}$ (Gaussienne)
   
   $$% latex2html id marker 16395\begin{array}{lll} \displaysty... ...''\right) ^{2}}{2}} & \mathrm{avec}\, x''=\alpha x' \end{array}$$
   $$% latex2html id marker 16397\begin{array}{ll} \displaystyl... ...\int\limits _{0}^{\infty }dt\, e^{-t}\\ & =2\pi \end{array}$$
   $\Rightarrow \boxed{\tilde{\psi }(k)=\frac{1}{\alpha }e^{-\frac{k^{2}}{2\alpha ^{2}}}}$
   $\Rightarrow$ La transformée d'une gaussienne est une gaussienne.
4. Dans les 3 cas: $\Delta x\sim \frac{1}{\alpha };\quad \Delta k\sim \alpha$
   
   $$\begin{array}{l} \psi (x)\xrightarrow {\alpha \rightarrow 0}... ...tarrow {\alpha \rightarrow 0}\sqrt{2\pi }\delta (k) \end{array}$$