mathml scratchpad

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

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

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

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

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

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