User:Ripe

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

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

$dx^2\; \backslash cdot\; ds^2\; =$

\frac {\int_{-\infty}^{\infty} {x^2f(x)f^*(x)dx} } { \int_{-\infty}^{\infty} {f(x)f^*(x)dx} }

\cdot

\frac {\int_{-\infty}^{\infty} {s^2f(s)f^*(s)dx} } { \int_{-\infty}^{\infty} {f(s)f^*(s)ds} }

$dx\; \backslash cdot\; ds\; \backslash ge\; \backslash frac\; \{1\}\; \{4\backslash pi\}$

$dx^2\; ds^2\; \backslash ge\; \backslash frac\; \{|\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; dx\; x\; f^*\; f\text{'}\; +\; x\; f\; f\text{'}^*\; |^2\}\; \{16\; \backslash pi\; (\backslash int\_\{-infty\}^\{\backslash infty\}\; f\; f^*\; dx)^2\; \}$

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

$dx^2\; ds^2\backslash ge\; \backslash frac\; \{|\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; dx\; x\; f^*\; f\text{'}\; +\; x\; f\; f\text{'}^*\; |^2\}\; \{16\; \backslash pi\; (\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; f\; f^*\; dx)^2\; \}$

$\backslash frac\; \{|\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; dx\; f\; f^*\; |\; ^2\}\; \{16\; \backslash pi\; (\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; dx\; f\; f^*)\; ^2\; \}\; \backslash sim\; \backslash frac\; \{1\}\{16\backslash pi\}\; \backslash le\; dx^2ds^2$

$\backslash frac\; \{1\}\; \{4\backslash pi\}\; \backslash le\; dx\; ds$

$\backslash int\_0^\backslash infty$

\Delta p \ge \frac{\hbar}{2}

Gabor function:

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