PX262 - C7a - the uncertainty principle

derivation

if $\hat{Q}$ and $\hat{R}$ are hermitian, then $(\hat{Q} \hat{R} + \hat{R} \hat{Q})$ and $i (\hat{Q} \hat{R} - \hat{R} \hat{Q})$ are also hermitian
this means that ${\hat{Q}}^{2}$ is also hermitian
the mean: $⟨ \hat{Q} ⟩ = \int ψ^{*} \hat{Q} ψ d τ$
the 'spread', or the standard deviation is: $Δ q^{2} = \int ψ^{*} (\hat{Q} - ⟨ \hat{Q} ⟩)^{2} ψ d τ = \int ψ^{*} {\hat{Q}}^{'} {\hat{Q}}^{'} ψ d τ$ where, ${\hat{Q}}^{'} = \hat{Q} - ⟨ \hat{Q} ⟩$
using the definition of hermitian operators: $Δ q^{2} = \int ({\hat{Q}}^{'} ψ) ({\hat{Q}}^{' *} ψ^{*}) d τ = \int | {\hat{Q}}^{'} ψ |^{2} d τ$
similarly, for $\hat{R} :$ $Δ r^{2} = \int | {\hat{R}}^{'} ψ |^{2} d τ$ where, ${\hat{R}}^{'} = \hat{R} - ⟨ \hat{R} ⟩$
multiplying the two spreads, and using schwarz's inequality: $Δ q^{2} Δ r^{2} = \int | {\hat{Q}}^{'} ψ |^{2} d τ \int | {\hat{R}}^{'} ψ |^{2} d τ \geq {| \int ({\hat{Q}}^{'} ψ^{*}) ({\hat{R}}^{'} ψ) d τ |}^{2}$

\begin{aligned} \int ({\hat{Q}}^{'} ψ^{*}) ({\hat{R}}^{'} ψ) d τ & = \int ψ^{*} {\hat{Q}}^{'} {\hat{R}}^{'} ψ d τ \\ = \frac{1}{2} \int ψ^{*} ({\hat{Q}}^{'} {\hat{R}}^{'} - {\hat{R}}^{'} {\hat{Q}}^{'}) ψ d τ + \frac{1}{2} \int ψ^{*} ({\hat{Q}}^{'} {\hat{R}}^{'} + {\hat{R}}^{'} {\hat{Q}}^{'}) ψ d τ \end{aligned}

the first integral is the imaginary part, and the second integral is the real part of the product

Δ q^{2} Δ r^{2} \geq (\frac{1}{4} {| \int ψ^{*} [{\hat{Q}}^{'}, {\hat{R}}^{'}] ψ d τ |}^{2} + {| \int ψ^{*} ({\hat{Q}}^{'} {\hat{R}}^{'} + {\hat{R}}^{'} {\hat{Q}}^{'}) ψ d τ |}^{2})

the second integral, which is real, will have a positive value for square, so: $Δ q^{2} Δ r^{2} \geq \frac{1}{4} {| \int ψ^{*} [{\hat{Q}}^{'}, {\hat{R}}^{'}] ψ d τ |}^{2}$
it can be shown that $[{\hat{Q}}^{'}, {\hat{R}}^{'}] = [\hat{Q}, \hat{R}]$ , so: $Δ q Δ r \geq \frac{1}{2} | ⟨ [\hat{Q}, \hat{R}] ⟩ |$
this is generalized heisenberg's uncertainty principle
for position and momentum operators, $[\hat{X}, {\hat{P}}_{x}] = i ℏ :$ $Δ x Δ p_{x} = \frac{ℏ}{2}$

the two quantities are measured on different copies of the system/different experiments
HUP does not tell how one variable is affected by the measurement of another, but rather how well given quantities can be measured or predicted
note: the generalized uncertainty principle is an inequality, so $[\hat{Q}, \hat{R}] = 0$ does not mean that $Δ q Δ r = 0$