D.19 The gen­er­al­ized un­cer­tainty re­la­tion­ship

This note de­rives the gen­er­al­ized un­cer­tainty re­la­tion­ship.

For brevity, de­fine $A'$ $\vphantom0\raisebox{1.5pt}{$=$}$ $A-\left\langle{A}\right\rangle$ and $B'$ $\vphantom0\raisebox{1.5pt}{$=$}$ $B-\langle{B}\rangle$, then the gen­eral ex­pres­sion for stan­dard de­vi­a­tion says

$$\sigma_A^2 \sigma_B^2 = \langle A'^2\rangle \langle B'^2\ra... ...le\Psi \vert A'^2\Psi\rangle \langle\Psi \vert B'^2\Psi\rangle$$

Her­mit­ian op­er­a­tors can be taken to the other side of in­ner prod­ucts, so

$$\sigma_A^2 \sigma_B^2 = \langle A'\Psi \vert A'\Psi\rangle \langle B'\Psi \vert B'\Psi\rangle$$

Now the Cauchy-Schwartz in­equal­ity says that for any $f$ and $g$,

$$\vert\langle f\vert g \rangle\vert \mathrel{\raisebox{-.7pt... ... \sqrt{\langle f\vert f\rangle}\sqrt{\langle g\vert g\rangle}$$

(See the no­ta­tions for more on this the­o­rem.) Us­ing the Cauchy-Schwartz in­equal­ity in re­versed or­der, you get

$$\sigma_A^2 \sigma_B^2 \mathrel{\raisebox{-1pt}{$\geqslant$}... ... \vert B'\Psi\rangle\vert^2 = \vert\langle A' B'\rangle\vert^2$$

Now by the de­f­i­n­i­tion of the in­ner prod­uct, the com­plex con­ju­gate of $\left\langle\vphantom{B'\Psi}A'\Psi\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{A'\Psi}B'\Psi\right\rangle$ is $\left\langle\vphantom{A'\Psi}B'\Psi\hspace{-\nulldelimiterspace}\hspace{.03em}\right.\!\left\vert\vphantom{B'\Psi}A'\Psi\right\rangle$, so the com­plex con­ju­gate of $\left\langle{A'B'}\right\rangle$ is $\left\langle{B'A'}\right\rangle$, and av­er­ag­ing a com­plex num­ber with mi­nus its com­plex con­ju­gate re­duces its size, since the real part av­er­ages away, so

$$\sigma_A^2 \sigma_B^2 \mathrel{\raisebox{-1pt}{$\geqslant$}... ...\frac{\langle A' B'\rangle-\langle B' A'\rangle}2\right\vert^2$$

The quan­tity in the top is the ex­pec­ta­tion value of the com­mu­ta­tor $[A',B']$. Writ­ing it out shows that $[A',B']$ $\vphantom0\raisebox{1.5pt}{$=$}$ $[A,B]$.