uncertainty relations
my biggest goal this semester is to know the physics I’m learning well, and one of the steps is to cement my understanding of important tools, procedures, and results. i’m also working on an 8.20 (sr) post but that’s second priority to 8.04/5 review for 8.06 this semester. general uncertainty principle schwarz inequality An important inequality you must know is the Schwarz inequality–$\langle a, a \rangle \langle b, b \rangle \geq |\langle a, b \rangle|^2$. Here we are using our complex inner product, which can yield complex values. In order to derive the inequality, first write $a = cb + a_\perp$, splitting $a$ into a part parallel to $b$ and a part perpendicular to $b$. In order for this to work, $c = \frac{\langle a, b\rangle}{\langle b,b\rangle}$. Think of it as taking the dot product of $a$ with the $\hat b$ direction. We can verify that $\langle a_\perp, b\rangle = \langle a - cb, b \rangle = \langle a,b \rangle - c \langle b,b \rangle = \langle a, b \rangle - \langle a, b \rangle = 0$ as desired. Then $$\begin{align*} \langle a,a\rangle \langle b,b \rangle &= \langle cb + a_\perp, cb + a_\perp \rangle \langle b,b \rangle \\ &= (|c|^2 \langle b,b \rangle + |a_\perp|^2)\langle b,b \rangle \\ &= |c|^2|b|^2 + |a_\perp|^2|b|^2 \\ &= |\langle a,b, \rangle|^2 + |a_\perp|^2|b|^2 \\ &\geq |\langle a,b, \rangle|^2 \end{align*}$$ as desired. ...