User:Ripe

${\displaystyle dx^{2}={\frac {\int _{-\infty }^{\infty }{x^{2}f(x)f^{*}(x)dx}}{\int _{-\infty }^{\infty }{f(x)f^{*}(x)dx}}}}$

${\displaystyle ds^{2}={\frac {\int _{-\infty }^{\infty }{s^{2}f(s)f^{*}(s)dx}}{\int _{-\infty }^{\infty }{f(s)f^{*}(s)ds}}}}$

${\displaystyle dx^{2}\cdot ds^{2}={\frac {\int _{-\infty }^{\infty }{x^{2}f(x)f^{*}(x)dx}}{\int _{-\infty }^{\infty }{f(x)f^{*}(x)dx}}}\cdot {\frac {\int _{-\infty }^{\infty }{s^{2}f(s)f^{*}(s)dx}}{\int _{-\infty }^{\infty }{f(s)f^{*}(s)ds}}}}$

${\displaystyle dx\cdot ds\geq {\frac {1}{4\pi }}}$

${\displaystyle dx^{2}ds^{2}\geq {\frac {|\int _{-\infty }^{\infty }dxxf^{*}f'+xff'^{*}|^{2}}{16\pi (\int _{-infty}^{\infty }ff^{*}dx)^{2}}}}$

integrate by parts with u=x and dv = d/dx:

${\displaystyle dx^{2}ds^{2}\geq {\frac {|\int _{-\infty }^{\infty }dxxf^{*}f'+xff'^{*}|^{2}}{16\pi (\int _{-\infty }^{\infty }ff^{*}dx)^{2}}}}$

${\displaystyle {\frac {|\int _{-\infty }^{\infty }dxff^{*}|^{2}}{16\pi (\int _{-\infty }^{\infty }dxff^{*})^{2}}}\sim {\frac {1}{16\pi }}\leq dx^{2}ds^{2}}$

${\displaystyle {\frac {1}{4\pi }}\leq dxds}$

${\displaystyle \int _{0}^{\infty }\Delta p\geq {\frac {\hbar }{2}}}$

Gabor function: ${\displaystyle G(t)\propto e^{[{\frac {-t^{2}}{2a^{2}}}+i(kt+\theta )]}}$