User:Ripe

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

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

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

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

$$ dx^2 ds^2 \ge \frac {|\int_{-\infty}^{\infty} dx x f^* f' + x f f'^* |^2} {16 \pi (\int_{-infty}^{\infty} f f^* dx)^2 } $$

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

$$ dx^2 ds^2\ge \frac {|\int_{-\infty}^{\infty} dx x f^* f' + x f f'^* |^2} {16 \pi (\int_{-\infty}^{\infty} f f^* dx)^2 } $$

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

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

$$\int_0^\infty \Delta p \ge \frac{\hbar}{2}$$

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