Examples

\begin{align}Et \geq \frac{\hbar}{2} \notag\end{align}

\begin{align}Et = \frac{\hbar}{2}(2n + 1) \notag\end{align}

\begin{align}I = \frac{E t}{\Delta E \Delta T} = \left(\frac{2}{\hbar}\right) Et = (2n+1) \geq 1 \text{ nat} = \log_2 (e)\text{ bits} \notag\end{align}

\begin{align}&n \text{ measurement locations on screen} \notag \\&I (\text{no measurement @ slit}) = h(\cos^2(x)) - h(\text{uniform}(x)) + \log(n)\notag \\&I (\text{measurement @ slit}) = \log(2) + \log(n) \notag \\&h (\text{uniform}(x)) - h(\cos^2(x)) =  \log(e/2) \notag \\&I (\text{measurement}) - I (\text{no measurement}) = \log(2) + \log(e/2) = 1 \notag\end{align}

\begin{align}\Delta E \Delta t = \frac{\hbar}{2} \notag\end{align}

\begin{align}I = \left(\frac{2}{\hbar}\right) E t = 1 \notag\end{align}

\begin{align}h(p(x)) = -\int p(x) \log(p(x)) dx \notag\end{align}

\begin{align}I = h(p(f)) + h(p(t)) = \frac{1}{2} \log(8\pi e \Delta f^2) + \frac{1}{2}\log(2 \pi e \Delta t^2) = 1 \notag\end{align}

\begin{align}S_t = \frac{\hbar}{2} \left[\begin{array}{cc}1 & 0 \\0 & 1 \\\end{array} \right] \notag\end{align}

\begin{align}|S| = \sqrt{|S_x|^2 + |S_y|^2 + |S_z|^2 + |S_t|^2 } = \hbar\left(s + \frac{1}{2}\right)\notag\end{align}

\begin{align}I_{\text{spin}} = \frac{2|S|}{\hbar} = 2s + 1\notag\end{align}

\begin{align}I_{S_x} = I_{S_y} = I_{S_z} = \int^\pi_0 \frac{1}{2} \sin (\theta) H_2 \left( \beta = \cos \left(\frac{\theta}{2}\right)^2 \right) d\theta = \frac{1}{2}\notag\end{align}

\begin{align}I_{S_t} = \int^{\delta t}_0 \frac{1}{\delta t} H_2 \left(\beta = \frac{t}{\delta t}\right) dt = \frac{1}{2}\notag\end{align}

\begin{align}I_{\text{spin}-\frac{1}{2}} = I_{S_x} + I_{S_y} + I_{S_z} + I_{S_t} = 2\notag\end{align}