Uncertainty Principle

The product of a function’s widths in the time domain ($\Delta t$) and the frequency domain (spectral width) ($\Delta \nu$) must be greater than ~1

• From the Scale Theorem: the shorter a Pulse in time, the broader the spectrum, and vice versa $⟹$ Cannot have a short pulse that is monochromatic
• Not inherent to Quantum Mechanics, but to Waves

$$\begin{array}{r}\Delta t=\frac{1}{f(0)}{\int }_{-\infty }^{\infty }|f(t)|dt\\ \Delta \omega =\frac{1}{F(0)}{\int }_{-\infty }^{\infty }|F(t)|dt\end{array}$$ $$\begin{array}{r}\Delta t=\frac{1}{f(0)}{\int }_{-\infty }^{\infty }|f(t)|dt\equiv \\ \Delta t\ge \frac{1}{f(0)}{\int }_{-\infty }^{\infty }f(t)dt\equiv \\ \Delta t\ge \frac{1}{f(0)}{\int }_{-\infty }^{\infty }f(t)(\exp (i\omega (0)))dt\equiv \\ \Delta t\ge \frac{F(0)}{f(0)}\end{array}$$ $$\begin{array}{r}\Delta \omega =\frac{1}{F(0)}{\int }_{-\infty }^{\infty }|F(\omega )|d\omega \equiv \\ \Delta \omega \ge \frac{1}{F(0)}{\int }_{-\infty }^{\infty }F(\omega )d\omega \equiv \\ \Delta \omega \ge \frac{1}{F(0)}{\int }_{-\infty }^{\infty }F(\omega )(\exp (i\omega (0)))d\omega \equiv \\ \Delta \omega \ge \frac{2\pi f(0)}{F(0)}\end{array}$$ $$\Delta \omega \Delta t\ge 2\pi \equiv \Delta \nu \Delta t\ge 1$$

Also:

$$\Delta {\omega }_{\text{rms}}\Delta t_{\text{rms}}\ge \frac{1}{2}$$

For functions of $x$ and $k$

$$\Delta k\Delta x\ge \frac{1}{2}$$

Heisenberg’s Uncertainty Principle for Momentum and Position

$$\Delta p\Delta x\ge \frac{\hslash }{2}$$

Heisenberg’s Uncertainy Principle for Energy and Time

$$\Delta E\Delta t\ge \frac{\hslash }{2}$$