███████████ █████████

The ███████████ █████████, ████ █████ as ██████████'s █████████████ █████████, is a ███████████ ███████ in ███████ █████████. It ██████ that there is a █████ to the █████████ with which ███████ █████ of ████████ ██████████, ████ as ████████ and ████████, can be ██████████████ █████. In █████ █████, the ████ ██████████ ███ ████████ is ████████, the ████ ██████████ the █████ ████████ can be █████.

████ ████████, the ███████████ █████████ is ███ of a ███████ of ████████████ ████████████ █████████ a ███████████ █████ to the ███████ of the ████████ of ███████ ███████ █████ of ████████████ on a ███████ ██████, ████ as ████████, █, and ████████, █. ████ ██████-█████████ are █████ as █████████████ █████████ or ███████████ █████████ █████████.

█████ ██████████ in ████ by ██████ █████████ ██████ ██████████, the ██████ ██████████ ████████ the ████████ █████████ of ████████ █_█ and the ████████ █████████ of ████████ █_█ was ███████ by █████ █████ ███████ █████ that ████ and by ███████ ████ in ████:

$\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}$

█████ $\hbar ={\frac {h}{2\pi }}$ is the ███████ ██████ ████████.

The ████████████████ ███████ ██████████ ███████████ █████████ █████ in ████ █████ █████ than ████████–████████. The ██████–████ ████████████ is ██████ used to ██████ ███████ █████ ████████ to ████████ ██████ ██████ but ███ ██████ ██████████ is ███████ with █████████ ██████ about the ██████ of ████. The █████ █████████ has been ████████ in ████████ ██████████; it ████ be ██████████ in ████ █████ of ███████████ ████████ ████████████.

████████–████████

It is █████ to ██████████ ███ the █████████ ███████ to ██████████ ████████████ ████████ ██████████ since it is █████████████ on the ███████████ ██████ that ██████ ██████████. ███ ███████████ ██████████ for ███████ ███████ █████ █████████ ████████████ for the ███████████ █████████. The ████ █████████ ███████ of the ███████████ █████████ is ████ ████████ █████████, but the ████ ████████ ██████ █████████ ███████ ██████████ it in a ███ that ███████████ ████ ██████. The ██████ of ██████████ ████████████ █████████ is ███████████ ████ for ██████ of ███████████ ███████ and is ██████████ for that of ███████████ ███████ .

██████████████, in ████ █████████, the ███████████ ████████ between ████████ and ████████ ██████ because the ███████████ of the ████████████ in the ███ █████████████ ███████████ █████ in ███████ █████ are ███████ ██████████ of ███ ███████ (█.█., ████████ and ████████ are █████████ █████████). A ███████ ████████ and ███ ███████ █████████ ██████ ████ be ███████ █████████ at the ████ ████. A ███████ ████████ between the █████████ of ███████ ██████████ ██████ in ███ ███████ █████████ by ███████ ████████, for ███████ in █████ █████: A ████ ████ is a █████ █████ at a ██████ █████████, █████ ███ ███████ █████████ █████ the █████ of the █████ ████ in the ████ ██████, which is a ██████████ ███████████ ████ ████. In ███████ █████████, the ███ ███ ██████ are that the ████████ of the ████████ █████ the ████ of a ██████ ████, and ████████ is ███ ███████ █████████, ███████ by the ██ ███████ ████████ █ = ██, █████ █ is the ██████████.

In ██████ █████████, the ████████████ ███████████ of ███████ █████████, ███ ████ of ███-█████████ ████-███████ █████████ ████████████ ███████████ are ███████ to ███████ ███████████ ██████. An ██████████ of an ██████████ ██████████ the █████ of the ████████████ for a ███████ ███████████ █████ (the ██████████). For ███████, if a ███████████ of an ██████████ A is █████████, ████ the ██████ is in a ██████████ ██████████ █ of that ██████████. ███████, the ██████████ ██████████ of the ██████████ A ████ not be an ██████████ of ███████ ██████████ █: If ██, ████ it ████ not have a ██████ ██████████ ███████████ for it, as the ██████ is not in an ██████████ of that ██████████.

█████████████

The ███████████ █████████ can be ██████████ using the ████████- and ████████-█████ █████████████ for ███ ████████ ████████ with ████ in ███ █████████.

The ████ █████████ the ████████-█████ ████████████, the ████ ██████ the ████████ is to be █████ with the ████████ ███████████ in that ██████, and ███████████████ the ████████-█████ ████████████ is ████ █████████ ██ the ████████ ████████ ██████████ the ████████ █████ have are ████ ██████████. ██████████, the ████ █████████ the ████████-█████ ████████████, the ████ ██████ the ████████ is to be █████ with █████ ██████ of ████████ ██████████ in that ██████, and ███████████████ the ████ █████████ the ████████-█████ ████████████, ██ the ████████ ███████████ the ████████ █████ ██████ are ████ ██████████. █████ █████████████ are ███████ ██████████ of ████ █████: ██████████████, the ███████████ █████████ █████████ the ████████████ between █████████ █████████ in the █████████.

████ █████████ ██████████████

█████████ to the ██ ███████ ██████████, █████ ██████ in the ████████ is ██████████ with a ████. ████ █████ ██████, from an ██████████ ████████ to █████, █████████ and on up to ███████ and beyond are ███████ to the ███████████ █████████.

The ████-███████████ ████ ████████ of a ██████-█████ █████ ████ of ██████████ █_█ or ████████ █_█ is $\psi (x)\propto e^{ik_{0}x}=e^{ip_{0}x/\hbar }~.$

The ████ ████ ██████ that this ██████ be ███████████ as a ███████████ ███████ █████████ ████████ in the █████ that the ███████████ of ███████ the ████████ between a and █ is $\operatorname {P} [a\leq X\leq b]=\int _{a}^{b}|\psi (x)|^{2}\,\mathrm {d} x~.$

In the ████ of the ██████-████ █████ ████, $|\psi (x)|^{2}$ is █ if $X=x$ and █ █████████. In █████ █████, the ████████ ████████ is █████████ █████████ in the █████ that it █████ be ███████████ ████████ along the ████ ██████.

On the █████ ████, ████████ a ████ ████████ that is a ███ of ████ █████, which ██ ███ █████ as $\psi (x)\propto \sum _{n}A_{n}e^{ip_{n}x/\hbar }~,$ █████ A_█ ██████████ the ████████ ████████████ of the ████ █_█ to the ███████ █████. The ███████ to the █████ ████ ███ with the ████████ of ████ █████ █████, the ████ ██████ can ██████ ████ █████████. ██ ███ ████ this a ████ ███████ to the █████████ █████, █████ the ████ ████████ is an ████████ over ███ ████████ █████ $\psi (x)={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }\varphi (p)\cdot e^{ipx/\hbar }\,dp~,$ with $\varphi (p)$ ████████████ the █████████ of █████ █████ and is ██████ the ████ ████████ in ████████ █████. In ████████████ █████, ██ ███ that $\varphi (p)$ is the ███████ █████████ of $\psi (x)$ and that █ and █ are █████████ █████████. ██████ ████████ ███ of █████ █████ █████ █████ at a ████, ██████ the ████████ has ██████ ████ ███████, ██████ ██████ a ███████ of █████ of ████ █████████ ███████.

███ ███ to ████████ the █████████ of the ████████ and ████████ is the ████████ █████████ █. Since $|\psi (x)|^{2}$ is a ███████████ ███████ ████████ for ████████, ██ █████████ ███ ████████ █████████.

The █████████ of the ████████ is ████████, █.█. ███████ █_█, by using ████ █████ █████, ███████ █████████ the █████████ of the ████████, █.█. █████████ █_█. ███████ ███ of ███████ this is that █_█ and █_█ have an ███████ ████████████ or are at █████ ███████ from below. This is the ███████████ █████████, the █████ █████ of which is the ███████ █████.

█████ of the ███████ ██████████ using ████ █████████

██ are ██████████ in the █████████ of ████████ and ████████, ███████ as $\sigma _{x}^{2}=\int _{-\infty }^{\infty }x^{2}\cdot |\psi (x)|^{2}\,dx-\left(\int _{-\infty }^{\infty }x\cdot |\psi (x)|^{2}\,dx\right)^{2}$ $\sigma _{p}^{2}=\int _{-\infty }^{\infty }p^{2}\cdot |\varphi (p)|^{2}\,dp-\left(\int _{-\infty }^{\infty }p\cdot |\varphi (p)|^{2}\,dp\right)^{2}~.$

Without ████ of ██████████, ██ will ██████ that the █████ ██████, which ████ ███████ to a █████ of the ██████ of ███ ███████████. (A ████ ███████ █████ that ████ not ████ this ██████████ is █████ below.) This █████ ██ the ███████ ████ $\sigma _{x}^{2}=\int _{-\infty }^{\infty }x^{2}\cdot |\psi (x)|^{2}\,dx$ $\sigma _{p}^{2}=\int _{-\infty }^{\infty }p^{2}\cdot |\varphi (p)|^{2}\,dp~.$

The ████████ $f(x)=x\cdot \psi (x)$ can be ███████████ as a ██████ in a ████████ █████. ██ can ██████ an █████ ███████ for a ████ of █████████ █(█) and █(█) in this ██████ █████: $\langle u\mid v\rangle =\int _{-\infty }^{\infty }u^{*}(x)\cdot v(x)\,dx,$ █████ the ████████ ███████ the ███████ █████████.

With this █████ ███████ ███████, ██ ████ that the ████████ for ████████ can be ███████ as $\sigma _{x}^{2}=\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=\langle f\mid f\rangle ~.$

██ can ██████ this for ████████ by ████████████ the ████████ ${\tilde {g}}(p)=p\cdot \varphi (p)$ as a ██████, but ██ can ████ ████ █████████ of the ████ that $\psi (x)$ and $\varphi (p)$ are ███████ ██████████ of ████ █████. ██ ████████ the ███████ ███████ █████████ through ███████████ by █████: ${\begin{aligned}g(x)&={\frac {1}{\sqrt {2\pi \hbar }}}\cdot \int _{-\infty }^{\infty }{\tilde {g}}(p)\cdot e^{ipx/\hbar }\,dp\\&={\frac {1}{\sqrt {2\pi \hbar }}}\int _{-\infty }^{\infty }p\cdot \varphi (p)\cdot e^{ipx/\hbar }\,dp\\&={\frac {1}{2\pi \hbar }}\int _{-\infty }^{\infty }\left[p\cdot \int _{-\infty }^{\infty }\psi (\chi )e^{-ip\chi /\hbar }\,d\chi \right]\cdot e^{ipx/\hbar }\,dp\\&={\frac {i}{2\pi }}\int _{-\infty }^{\infty }\left[{\cancel {\left.\psi (\chi )e^{-ip\chi /\hbar }\right|_{-\infty }^{\infty }}}-\int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}e^{-ip\chi /\hbar }\,d\chi \right]\cdot e^{ipx/\hbar }\,dp\\&=-i\int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}\left[{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\,e^{ip(x-\chi )/\hbar }\,dp\right]\,d\chi \\&=-i\int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}\left[\delta \left({\frac {x-\chi }{\hbar }}\right)\right]\,d\chi \\&=-i\hbar \int _{-\infty }^{\infty }{\frac {d\psi (\chi )}{d\chi }}\left[\delta \left(x-\chi \right)\right]\,d\chi \\&=-i\hbar {\frac {d\psi (x)}{dx}}\\&=\left(-i\hbar {\frac {d}{dx}}\right)\cdot \psi (x),\end{aligned}}$ █████ $v={\frac {\hbar }{-ip}}e^{-ip\chi /\hbar }$ in the ███████████ by █████, the █████████ ████ ████████ because the ████ ████████ ████████ at ████ ██████████ and $|e^{-ip\chi /\hbar }|=1$ , and ████ use the █████ █████ ████████ which is █████ because ${\dfrac {d\psi (\chi )}{d\chi }}$ ████ not ██████ on █ .

The ████ ${\textstyle -i\hbar {\frac {d}{dx}}}$ is ██████ the ████████ ████████ in ████████ █████. ████████ ██████████'s ███████, ██ ███ that the ████████ for ████████ can be ███████ as $\sigma _{p}^{2}=\int _{-\infty }^{\infty }|{\tilde {g}}(p)|^{2}\,dp=\int _{-\infty }^{\infty }|g(x)|^{2}\,dx=\langle g\mid g\rangle .$

The ██████–███████ ██████████ ███████ that $\sigma _{x}^{2}\sigma _{p}^{2}=\langle f\mid f\rangle \cdot \langle g\mid g\rangle \geq |\langle f\mid g\rangle |^{2}~.$

The ███████ ███████ of ███ ███████ ██████ █ can be █████████ as $|z|^{2}={\Big (}{\text{Re}}(z){\Big )}^{2}+{\Big (}{\text{Im}}(z){\Big )}^{2}\geq {\Big (}{\text{Im}}(z){\Big )}^{2}=\left({\frac {z-z^{\ast }}{2i}}\right)^{2}.$ ██ ███ $z=\langle f|g\rangle$ and $z^{*}=\langle g\mid f\rangle$ and ██████████ █████ into the ████████ above to ███ $|\langle f\mid g\rangle |^{2}\geq \left({\frac {\langle f\mid g\rangle -\langle g\mid f\rangle }{2i}}\right)^{2}~.$

███ that ███████ is to ████████ █████ █████ ████████.

${\begin{aligned}\langle f\mid g\rangle -\langle g\mid f\rangle &=\int _{-\infty }^{\infty }\psi ^{*}(x)\,x\cdot \left(-i\hbar {\frac {d}{dx}}\right)\,\psi (x)\,dx-\int _{-\infty }^{\infty }\psi ^{*}(x)\,\left(-i\hbar {\frac {d}{dx}}\right)\cdot x\,\psi (x)\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }\psi ^{*}(x)\left[\left(-x\cdot {\frac {d\psi (x)}{dx}}\right)+{\frac {d(x\psi (x))}{dx}}\right]\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }\psi ^{*}(x)\left[\left(-x\cdot {\frac {d\psi (x)}{dx}}\right)+\psi (x)+\left(x\cdot {\frac {d\psi (x)}{dx}}\right)\right]\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }\psi ^{*}(x)\psi (x)\,dx\\&=i\hbar \cdot \int _{-\infty }^{\infty }|\psi (x)|^{2}\,dx\\&=i\hbar \end{aligned}}$

████████ this into the above ████████████, ██ ███ $\sigma _{x}^{2}\sigma _{p}^{2}\geq |\langle f\mid g\rangle |^{2}\geq \left({\frac {\langle f\mid g\rangle -\langle g\mid f\rangle }{2i}}\right)^{2}=\left({\frac {i\hbar }{2i}}\right)^{2}={\frac {\hbar ^{2}}{4}}$ and ██████ the ██████ ████ $\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}~.$

with ████████ if and ████ if █ and █ are ████████ █████████. ████ that the ████ ███████ ████████ in this █████ was that $\psi (x)$ and $\varphi (p)$ are ████ █████████ for ████████ and ████████, which are ███████ ██████████ of ████ █████. A ███████ ██████ █████ ████ for ███ ████ of █████████ █████████.

██████ █████████ ██████████████

In ██████ █████████, ███████████ ████ as ████████ and ████████ are ███████████ by ████-███████ █████████. ████ considering █████ of ███████████, an █████████ ████████ is the ██████████. For a ████ of █████████ Â and ${\hat {B}}$ , ███ ███████ █████ ██████████ as $[{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}.$ In the ████ of ████████ and ████████, the ██████████ is the █████████ ███████████ ████████ $[{\hat {x}},{\hat {p}}]=i\hbar .$

The ████████ ███████ of the ███-█████████████ can be ██████████ by considering the ██████ of the ██████████ on ████████ and ████████ ███████████. ███ $|\psi \rangle$ be a █████ ██████████ of ████████ with a ████████ ██████████ █_█. By ██████████, this █████ that ${\hat {x}}|\psi \rangle =x_{0}|\psi \rangle .$ ████████ the ██████████ to $|\psi \rangle$ ██████ $[{\hat {x}},{\hat {p}}]|\psi \rangle =({\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}})|\psi \rangle =({\hat {x}}-x_{0}{\hat {I}}){\hat {p}}\,|\psi \rangle =i\hbar |\psi \rangle ,$ █████ █ is the ████████ ████████.

███████, for the ████ of █████ by █████████████, that $|\psi \rangle$ is ████ a █████ ██████████ of ████████, with ████████ ██████████ █_█. If this were ████, ████ ███ █████ █████ $({\hat {x}}-x_{0}{\hat {I}}){\hat {p}}\,|\psi \rangle =({\hat {x}}-x_{0}{\hat {I}})p_{0}\,|\psi \rangle =(x_{0}{\hat {I}}-x_{0}{\hat {I}})p_{0}\,|\psi \rangle =0.$ On the █████ ████, the above █████████ ███████████ ████████ ████████ that $[{\hat {x}},{\hat {p}}]|\psi \rangle =i\hbar |\psi \rangle \neq 0.$ This ███████ that ██ ███████ █████ can ██████████████ be ████ a ████████ and a ████████ ██████████.

████ a █████ is ████████, it is █████████ onto an ██████████ in the █████ of the ████████ ██████████. For ███████, if a ████████'s ████████ is ████████, ████ the █████ ███████ to a ████████ ██████████. This █████ that the █████ is not a ████████ ██████████, ███████, but ██████ it can be ███████████ as a ███ of ████████ ████████ █████ ███████████. In █████ █████, the ████████ ████ be ████ ███████. This █████████ ███ be ██████████ by the ████████ ██████████, $\sigma _{x}={\sqrt {\langle {\hat {x}}^{2}\rangle -\langle {\hat {x}}\rangle ^{2}}}$ $\sigma _{p}={\sqrt {\langle {\hat {p}}^{2}\rangle -\langle {\hat {p}}\rangle ^{2}}}.$

As in the ████ █████████ ██████████████ above, ███ ████ a ████████ between the ██████████ ██████████ of the ███, ██████████ by the ███████████ █████████.

███████ ████████ ██████████ ██████████ ██████

████████ a ███-███████████ ███████ ████████ ██████████. It is ████████ to ███████ the ████████ and ████████ █████████ in █████ of the ████████ and ████████████ █████████: ${\hat {x}}={\sqrt {\frac {\hbar }{2m\omega }}}(a+a^{\dagger })$ ${\hat {p}}=i{\sqrt {\frac {m\omega \hbar }{2}}}(a^{\dagger }-a).$

Using the ████████ █████ for ████████ and ████████████ █████████ on the ██████ ███████████, $a^{\dagger }|n\rangle ={\sqrt {n+1}}|n+1\rangle$ $a|n\rangle ={\sqrt {n}}|n-1\rangle ,$ the █████████ ███ be ████████ ████████, $\sigma _{x}^{2}={\frac {\hbar }{m\omega }}\left(n+{\frac {1}{2}}\right)$ $\sigma _{p}^{2}=\hbar m\omega \left(n+{\frac {1}{2}}\right)\,.$ The ███████ of █████ ████████ ██████████ is ████ $\sigma _{x}\sigma _{p}=\hbar \left(n+{\frac {1}{2}}\right)\geq {\frac {\hbar }{2}}.~$

In ██████████, the above ███████ █████ is █████████ for the ██████ █████ █=█, for which the ███████████ ███████ is ████ the ██████ ████████████.

███████ ████████ ███████████ with ████████ ███████ █████████

In a ███████ ████████ ██████████ of ██████████████ ███████ █████████ █, █████ a █████ that is ██████ from the ██████ of the █████████ by some ████████████ █_█ as $\psi (x)=\left({\frac {m\Omega }{\pi \hbar }}\right)^{1/4}\exp {\left(-{\frac {m\Omega (x-x_{0})^{2}}{2\hbar }}\right)},$ █████ █ █████████ the █████ of the ███████ █████ but ████ not be the ████ as █. Through ███████████ over the ██████████, ██ can █████ for the ████ ████-█████████ ████████. After ████ ████████████, the ███████████ █████████ ██████ to $|\Psi (x,t)|^{2}\sim {\mathcal {N}}\left(x_{0}\cos {(\omega t)},{\frac {\hbar }{2m\Omega }}\left(\cos ^{2}(\omega t)+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right)$ $|\Phi (p,t)|^{2}\sim {\mathcal {N}}\left(-mx_{0}\omega \sin(\omega t),{\frac {\hbar m\Omega }{2}}\left(\cos ^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin ^{2}{(\omega t)}\right)\right),$ █████ ██ have used the ████████ ${\mathcal {N}}(\mu ,\sigma ^{2})$ to ██████ a ██████ ████████████ of ████ █ and ████████ █^█. ███████ the █████████ above and ████████ █████████████ ██████████, ██ can █████ the ███████ of the ████████ ██████████ as ${\begin{aligned}\sigma _{x}\sigma _{p}&={\frac {\hbar }{2}}{\sqrt {\left(\cos ^{2}{(\omega t)}+{\frac {\Omega ^{2}}{\omega ^{2}}}\sin ^{2}{(\omega t)}\right)\left(\cos ^{2}{(\omega t)}+{\frac {\omega ^{2}}{\Omega ^{2}}}\sin ^{2}{(\omega t)}\right)}}\\&={\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-1\right)\cos {(4\omega t)}}}\end{aligned}}$

From the █████████ ${\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\geq 2,\quad |\cos(4\omega t)|\leq 1,$ ██ can ████████ the following (the █████ ████ ████████ █████ ████ ████ █ = █): $\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{4}}{\sqrt {3+{\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-\left({\frac {1}{2}}\left({\frac {\Omega ^{2}}{\omega ^{2}}}+{\frac {\omega ^{2}}{\Omega ^{2}}}\right)-1\right)}}={\frac {\hbar }{2}}.$

████████ ██████

A ████████ █████ is a █████ ██████████ of the ████████████ ████████, ${\hat {a}}|\alpha \rangle =\alpha |\alpha \rangle ,$ which ███ be ███████████ in █████ of ████ ██████ as $|\alpha \rangle =e^{-{|\alpha |^{2} \over 2}}\sum _{n=0}^{\infty }{\alpha ^{n} \over {\sqrt {n!}}}|n\rangle$

In the ███████ █████ the ████████ █████ is a ███████ ████████ in a ███████ ████████ ██████████, the ████████ and ████████ █████████ ███ be █████████ in █████ of the ████████████ █████████ in the ████ ████████ above and used to █████████ the █████████, $\sigma _{x}^{2}={\frac {\hbar }{2m\omega }},$ $\sigma _{p}^{2}={\frac {\hbar m\omega }{2}}.$ █████████, █████ ████████ █████ █████████ the ███████ █████ $\sigma _{x}\sigma _{p}={\sqrt {\frac {\hbar }{2m\omega }}}\,{\sqrt {\frac {\hbar m\omega }{2}}}={\frac {\hbar }{2}}.$ with ████████ and ████████ ████ ████████████ an ██████ ${\textstyle {\sqrt {\hbar /2}}}$ in a "████████" ███. ████████, █████ ████████ ████████ █████ ████ █████████ the ███████ █████ ████████ the ██████████ █████████████ of ████████ and ████████ ████ not be ████████ in ███████.

████████ in a ███

████████ a ████████ in a ███-███████████ ███ of ██████ $█$ . The ██████████████ in ████████ and ████████ █████ are $\psi _{n}(x,t)={\begin{cases}A\sin(k_{n}x)\mathrm {e} ^{-\mathrm {i} \omega _{n}t},&0<x<L,\\0,&{\text{otherwise,}}\end{cases}}$ and $\varphi _{n}(p,t)={\sqrt {\frac {\pi L}{\hbar }}}\,\,{\frac {n\left(1-(-1)^{n}e^{-ikL}\right)e^{-i\omega _{n}t}}{\pi ^{2}n^{2}-k^{2}L^{2}}},$ █████ ${\textstyle \omega _{n}={\frac {\pi ^{2}\hbar n^{2}}{8L^{2}m}}}$ and ██ have used the ██ ███████ ████████ $p=\hbar k$ . The █████████ of $█$ and $█$ can be ██████████ ██████████: $\sigma _{x}^{2}={\frac {L^{2}}{12}}\left(1-{\frac {6}{n^{2}\pi ^{2}}}\right)$ $\sigma _{p}^{2}=\left({\frac {\hbar n\pi }{L}}\right)^{2}.$

The ███████ of the ████████ ██████████ is █████████ $\sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}.$ For ███ $n=1,\,2,\,3,\,\ldots$ , the ████████ ${\textstyle {\sqrt {{\frac {n^{2}\pi ^{2}}{3}}-2}}}$ is ███████ than █, ██ the ███████████ █████████ is █████ ████████. For █████████ ████████████, the ████████ █████ ██████ ████ $n=1$ , in which ████ $\sigma _{x}\sigma _{p}={\frac {\hbar }{2}}{\sqrt {{\frac {\pi ^{2}}{3}}-2}}\approx 0.568\hbar >{\frac {\hbar }{2}}.$

████████ ████████

██████ a ████████ █████████ has a ████████ █████ ████ ████████ █████████ by a ██████ ████████████ around some ████████ ████████ █_█ █████████ to $\varphi (p)=\left({\frac {x_{0}}{\hbar {\sqrt {\pi }}}}\right)^{1/2}\exp \left({\frac {-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}\right),$ █████ ██ have ██████████ a █████████ █████ ${\textstyle x_{0}={\sqrt {\hbar /m\omega _{0}}}}$ , with $\omega _{0}>0$ ██████████ the █████ of the ████████████—██. █████████████████████. If the █████ is ███████ to ██████ in ████ █████, ████ the ████-█████████ ████████ and ████████ █████ ████ █████████ are $\Phi (p,t)=\left({\frac {x_{0}}{\hbar {\sqrt {\pi }}}}\right)^{1/2}\exp \left({\frac {-x_{0}^{2}(p-p_{0})^{2}}{2\hbar ^{2}}}-{\frac {ip^{2}t}{2m\hbar }}\right),$ $\Psi (x,t)=\left({\frac {1}{x_{0}{\sqrt {\pi }}}}\right)^{1/2}{\frac {e^{-x_{0}^{2}p_{0}^{2}/2\hbar ^{2}}}{\sqrt {1+i\omega _{0}t}}}\,\exp \left(-{\frac {(x-ix_{0}^{2}p_{0}/\hbar )^{2}}{2x_{0}^{2}(1+i\omega _{0}t)}}\right).$

Since $\langle p(t)\rangle =p_{0}$ and $\sigma _{p}(t)=\hbar /({\sqrt {2}}x_{0})$ , this can be ███████████ as a ████████ ██████ along with ████████ ████████ at ███████████ ████ █████████. On the █████ ████, the ████████ █████████ of the ████████ is $\sigma _{x}={\frac {x_{0}}{\sqrt {2}}}{\sqrt {1+\omega _{0}^{2}t^{2}}}$ ████ that the ███████████ ███████ can ████ ████████ with ████ as $\sigma _{x}(t)\sigma _{p}(t)={\frac {\hbar }{2}}{\sqrt {1+\omega _{0}^{2}t^{2}}}$

████████████ █████████

████████ with ███████'s ██████████ of ████████-████████ ███████████, ██████ █████ █████████ █████████ a ███████████ for █████████ █████████ █████████ ${\hat {\mathcal {O}}}$ █████████ in █████ of █████ ████████ █████████ $\sigma _{\mathcal {O}}={\sqrt {\langle {\hat {\mathcal {O}}}^{2}\rangle -\langle {\hat {\mathcal {O}}}\rangle ^{2}}},$ █████ the ████████ $\langle {\hat {\mathcal {O}}}\rangle$ ████████ an ███████████ █████ of the ██████████ ███████████ by ████████ ${\hat {\mathcal {O}}}$ . For a ████ of █████████ ${\hat {A}}$ and ${\hat {B}}$ , ██████ █████ ██████████ as $[{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}},$ and the █████████ ███████████ ████████ is █████ by $\sigma _{A}\sigma _{B}\geq \left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|={\frac {1}{2}}\left|\langle [{\hat {A}},{\hat {B}}]\rangle \right|.$

█████ ███████████ ██████ ███ to █████ for ███████████ between the █████████, ██████ a ████████ ██████████, █████ as the █████████–███████████ ███████████ ████████,

$\sigma _{A}^{2}\sigma _{B}^{2}\geq \left|{\frac {1}{2}}\langle \{{\hat {A}},{\hat {B}}\}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle \right|^{2}+\left|{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle \right|^{2},$

█████ the ██████████████, $\{{\hat {A}},{\hat {B}}\}={\hat {A}}{\hat {B}}+{\hat {B}}{\hat {A}}$ is used.

█████ of the ███████████ ███████████ ████████

The ██████████ █████ ████ ████████████ and ██████ off of █████ █████ in █████████, ███████████ and ████████ █████████ ████ as █████████. For ███ █████████ ████████ ${\hat {A}}$ , █████ upon the ██████████ of ████████, ██ have $\sigma _{A}^{2}=\langle ({\hat {A}}-\langle {\hat {A}}\rangle )\Psi |({\hat {A}}-\langle {\hat {A}}\rangle )\Psi \rangle .$ ██ ███ $|f\rangle =|({\hat {A}}-\langle {\hat {A}}\rangle )\Psi \rangle$ and ████ $\sigma _{A}^{2}=\langle f\mid f\rangle \,.$

█████████, for ███ █████ █████████ ████████ ${\hat {B}}$ in the ████ █████ $\sigma _{B}^{2}=\langle ({\hat {B}}-\langle {\hat {B}}\rangle )\Psi |({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle =\langle g\mid g\rangle$ for $|g\rangle =|({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle .$

The ███████ of the ███ ██████████ can ████ be █████████ as

In █████ to ██████ the ███ ███████ $|f\rangle$ and $|g\rangle$ , ██ use the ██████–███████ ██████████ which is ███████ as $\langle f\mid f\rangle \langle g\mid g\rangle \geq |\langle f\mid g\rangle |^{2},$ and ████ ████████ (█) can be ███████ as

Since $\langle f\mid g\rangle$ is in ███████ a ███████ ██████, ██ use the ████ that the ███████ ███████ of ███ ███████ ██████ $█$ is ███████ as $|z|^{2}=zz^{*}$ , █████ $z^{*}$ is the ███████ █████████ of $█$ . The ███████ ███████ can ████ be █████████ as

██ ███ $z=\langle f\mid g\rangle$ and $z^{*}=\langle g\mid f\rangle$ and ██████████ █████ into the ████████ above to ███

The █████ ███████ $\langle f\mid g\rangle$ is ███████ out ██████████ as $\langle f\mid g\rangle =\langle ({\hat {A}}-\langle {\hat {A}}\rangle )\Psi |({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle ,$ and using the ████ that ${\hat {A}}$ and ${\hat {B}}$ are █████████ █████████, ██ ████ ${\begin{aligned}\langle f\mid g\rangle &=\langle \Psi |({\hat {A}}-\langle {\hat {A}}\rangle )({\hat {B}}-\langle {\hat {B}}\rangle )\Psi \rangle \\[4pt]&=\langle \Psi \mid ({\hat {A}}{\hat {B}}-{\hat {A}}\langle {\hat {B}}\rangle -{\hat {B}}\langle {\hat {A}}\rangle +\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle )\Psi \rangle \\[4pt]&=\langle \Psi \mid {\hat {A}}{\hat {B}}\Psi \rangle -\langle \Psi \mid {\hat {A}}\langle {\hat {B}}\rangle \Psi \rangle -\langle \Psi \mid {\hat {B}}\langle {\hat {A}}\rangle \Psi \rangle +\langle \Psi \mid \langle {\hat {A}}\rangle \langle {\hat {B}}\rangle \Psi \rangle \\[4pt]&=\langle {\hat {A}}{\hat {B}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle +\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle \\[4pt]&=\langle {\hat {A}}{\hat {B}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle .\end{aligned}}$

█████████ it can be █████ that $\langle g\mid f\rangle =\langle {\hat {B}}{\hat {A}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle .$

████, ██ have $\langle f\mid g\rangle -\langle g\mid f\rangle =\langle {\hat {A}}{\hat {B}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle -\langle {\hat {B}}{\hat {A}}\rangle +\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle =\langle [{\hat {A}},{\hat {B}}]\rangle$ and $\langle f\mid g\rangle +\langle g\mid f\rangle =\langle {\hat {A}}{\hat {B}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle +\langle {\hat {B}}{\hat {A}}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle =\langle \{{\hat {A}},{\hat {B}}\}\rangle -2\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle .$

██ ███ ██████████ the above ███ █████████ above ████ into ██. (█) and ███ $|\langle f\mid g\rangle |^{2}={\Big (}{\frac {1}{2}}\langle \{{\hat {A}},{\hat {B}}\}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle {\Big )}^{2}+{\Big (}{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle {\Big )}^{2}\,.$

████████████ the above into ████████ (█) ██ ███ the ███████████ ███████████ ████████ $\sigma _{A}\sigma _{B}\geq {\sqrt {{\Big (}{\frac {1}{2}}\langle \{{\hat {A}},{\hat {B}}\}\rangle -\langle {\hat {A}}\rangle \langle {\hat {B}}\rangle {\Big )}^{2}+{\Big (}{\frac {1}{2i}}\langle [{\hat {A}},{\hat {B}}]\rangle {\Big )}^{2}}}.$

This █████ has an █████ ███████ to the ███████ of the █████████ ████████. For the █████ to ████ █████, the ██████ ${\hat {B}}|\Psi \rangle$ has to be in the ██████ of the █████████ ████████ ${\hat {A}}$ , which is not ██████ the ████. In ████, the █████████ ███████████ ████████ is █████ if ${\hat {A}}$ is an █████ ████████ and ${\hat {B}}$ is the ██████████ with ███████ to this ████████. In this ███████, the ██████████ is a ███████ ████████—████ as in the ██████████ ███████████ ████████—and ███ there are ██████ █████ the ███████ of the █████████████ is ████. (███ the ██████████████ ███████ below.) This █████ can be ████████ by using a ███████████ ██████ for the █████, or by ███████ with an █████████████ ███████ of the █████████ ███████████ █████████.

████ that in the ███████ ████ of the █████████–███████████ ███████████ ████████, there is ██ ████ to ██████ that the █████████ ${\hat {A}}$ and ${\hat {B}}$ are ████-███████ █████████. It ████████ to ██████ that they are ██████ █████████ █████████. (The ███████████ between █████ ███ ███████ is █████████ ███████ over in the ███████ ██████████, █████ the ████ █████████ is used for ██████ or ████ ███████ of █████████. ███ ███████ █ of ████'s ████ for a ████████ ██████████ of this █████████ but █████████ ███████████.)

█████ █████

In the █████ █████ ███████████ of ███████ █████████, the █████████–███████████ ████████ ███████ from a ██████████ █████████ on a ████ ████-██████ ████████. █████ a ██████ ████████ $W(x,p)$ with ████ ███████ ★ and a ████████ █, the following is █████████ ████: $\langle f^{*}\star f\rangle =\int (f^{*}\star f)\,W(x,p)\,dx\,dp\geq 0~.$

████████ $f=a+bx+cp$ , ██ ██████ at $\langle f^{*}\star f\rangle ={\begin{bmatrix}a^{*}&b^{*}&c^{*}\end{bmatrix}}{\begin{bmatrix}1&\langle x\rangle &\langle p\rangle \\\langle x\rangle &\langle x\star x\rangle &\langle x\star p\rangle \\\langle p\rangle &\langle p\star x\rangle &\langle p\star p\rangle \end{bmatrix}}{\begin{bmatrix}a\\b\\c\end{bmatrix}}\geq 0~.$

Since this ██████████ █████████ is ████ for ███ a, █, and █, it ███████ that ███ the ███████████ of the ██████ are ███-████████.

The ███-████████ ███████████ ████ █████ a █████████████ ███-██████████ █████████ on the ███████████, $\det {\begin{bmatrix}1&\langle x\rangle &\langle p\rangle \\\langle x\rangle &\langle x\star x\rangle &\langle x\star p\rangle \\\langle p\rangle &\langle p\star x\rangle &\langle p\star p\rangle \end{bmatrix}}=\det {\begin{bmatrix}1&\langle x\rangle &\langle p\rangle \\\langle x\rangle &\langle x^{2}\rangle &\left\langle xp+{\frac {i\hbar }{2}}\right\rangle \\\langle p\rangle &\left\langle xp-{\frac {i\hbar }{2}}\right\rangle &\langle p^{2}\rangle \end{bmatrix}}\geq 0~,$ or, ██████████, after █████████ ████████████, $\sigma _{x}^{2}\sigma _{p}^{2}=\left(\langle x^{2}\rangle -\langle x\rangle ^{2}\right)\left(\langle p^{2}\rangle -\langle p\rangle ^{2}\right)\geq \left(\langle xp\rangle -\langle x\rangle \langle p\rangle \right)^{2}+{\frac {\hbar ^{2}}{4}}~.$

████████

Since the █████████ and ███████████ █████████ are for ███████ █████████, the █████████ can be ███████ to ███ ███ ███████████ to ██████ ████████ ███████████ █████████. A ███ of the ████ ██████ █████████ █████ in the ██████████ are █████ below.

████████–██████ ████████ ███████████ ████████: for the ████████ and ██████ ████████ █████████, the █████████ ███████████ ████████ $[{\hat {x}},{\hat {p}}]=i\hbar$ ███████ the ███████ ██████████ from above: $\sigma _{x}\sigma _{p}\geq {\frac {\hbar }{2}}.$
███████ ████████ ███████████ ████████: For ███ ██████████ ██████████ of the █████ ███████ ████████ ████████ of an ██████: $\sigma _{J_{i}}\sigma _{J_{j}}\geq {\frac {\hbar }{2}}{\big |}\langle J_{k}\rangle {\big |},$ █████ █, █, █ are ████████, and █_█ ███████ ███████ ████████ along the █_█ ████. This ████████ ███████ that ██████ ███ █████ ██████████ ██████ ████████, ████ a ██████ █████████ of a ██████'s ███████ ████████ can be ███████ with █████████ █████████, ████████ the █████████ ████████ to an ████████ (████████ or ████████) █████. ████████, for $[J_{x},J_{y}]=i\hbar \varepsilon _{xyz}J_{z}$ , a ██████ ${\hat {A}}=J_{x}$ , ${\hat {B}}=J_{y}$ , in ███████ ████████ ██████████, █ = |█, █⟩, ██████ the ███████ █████████ (███████ ████████ ███████, $\langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle$ ) from below and ████ ██████ ██████ ███████████ ████ as █(█ + █) ≥ █(█ + █), and █████ █ ≥ █, among ██████.

For the ██████ of █████████ in a ██████████████ and the █████ of ███ ████████–██████ █████ █████████ $\Delta N\,\Delta \varphi \geq 1.$

███████████

The ██████████ of the █████████ ██████████ for █████████ ${\hat {A}}$ and ${\hat {B}}$ ████████ ${\hat {A}}{\hat {B}}\psi$ and ${\hat {B}}{\hat {A}}\psi$ to be ███████. There are ███████ ███████ █████ █████ ██████████ are not █████. ███ ███████ is a ███████ ████████ on a ████, █████ the ████ ████████ ███████ on an ███████ ████████ $\theta$ in the ████████ $[0,2\pi ]$ . ██████ "████████" and "████████" █████████ ${\hat {A}}$ and ${\hat {B}}$ by ${\hat {A}}\psi (\theta )=\theta \psi (\theta ),\quad \theta \in [0,2\pi ],$ and ${\hat {B}}\psi =-i\hbar {\frac {d\psi }{d\theta }},$ with ████████ ████████ ██████████ on ${\hat {B}}$ . The ██████████ of ${\hat {A}}$ ███████ the $\theta$ █████ from █ to $2\pi$ . █████ █████████ ███████ the █████ ███████████ █████████ for ████████ and ████████ █████████, $[{\hat {A}},{\hat {B}}]=i\hbar$ . ████ █████████, ${\hat {A}}{\hat {B}}\psi -{\hat {B}}{\hat {A}}\psi =i\hbar \psi$ ████████ ████ ${\hat {A}}{\hat {B}}\psi$ and ${\hat {B}}{\hat {A}}\psi$ are ███████, and the █████ of ████ $\psi$ is a █████ ████████ of the ███████ ███████ █████.

███ ███ $\psi$ be ███ of the ███████████ of ${\hat {B}}$ , which are █████ by $\psi (\theta )=e^{2\pi in\theta }$ . █████ ██████ are ████████████, unlike the ███████████ of the ████████ ████████ on the ████. ████ the ████████ ${\hat {A}}$ is ███████, since $\theta$ ██████ over a ███████ ████████. ████, in the █████ $\psi$ , the ███████████ of $█$ is ████ and the ███████████ of $A$ is ██████, ██ that $\sigma _{A}\sigma _{B}=0.$ The █████████ ███████████ █████████ ████ not █████ in this ████: $\psi$ is not in the ██████ of the ████████ ${\hat {B}}{\hat {A}}$ , since ██████████████ by $\theta$ ████████ the ████████ ████████ ██████████ ███████ on ${\hat {B}}$ .

For the █████ ████████ and ████████ █████████ ${\hat {X}}$ and ${\hat {P}}$ on the ████ ████, ██ ████ ███████████████ can █████. As ████ as $\sigma _{x}$ and $\sigma _{p}$ are ███████ in the █████ $\psi$ , the ██████████ ███████████ █████████ █████, ████ if $\psi$ █████ to be in the ██████ of ${\hat {X}}{\hat {P}}$ or of ${\hat {P}}{\hat {X}}$ .

█████ ██████

The █████████–███████████ ███████████ can be ████████ ██████ that it ████ ████ for ███ ██████████ $\varrho _{k}$ in ███ █████████████ of the ███████ ██████ █████ as $\varrho =\sum _{k}p_{k}\varrho _{k}.$ ████, for the █████████████ $p_{k}\geq 0$ and $\sum _{k}p_{k}=1$ ████. ████, using the ████████ $\sum _{k}a_{k}\sum _{k}b_{k}\geq \left(\sum _{k}{\sqrt {a_{k}b_{k}}}\right)^{2}$ for $a_{k},b_{k}\geq 0$ , it ███████ that $\sigma _{A}^{2}\sigma _{B}^{2}\geq \left[\sum _{k}p_{k}L(\varrho _{k})\right]^{2},$ █████ the ████████ in the █████ is ███████ $L(\varrho )={\sqrt {\left|{\frac {1}{2}}\operatorname {tr} (\rho \{A,B\})-\operatorname {tr} (\rho A)\operatorname {tr} (\rho B)\right|^{2}+\left|{\frac {1}{2i}}\operatorname {tr} (\rho [A,B])\right|^{2}}}.$ The above ████████ ████ █████ has a █████ ██████ than that of the ████████ █████████–███████████ ███████████ ████████. ████, ██ ████ to █████████ the █████ of the █████████–███████████ ███████████ for the █████ ██████████ of the ███████ █████ ██████ than for the ███████ █████, and ███████ an ███████ of █████ ██████ █████. The following ██████████ is ████████ than the █████████–███████████ ███████████ ████████ $\sigma _{A}^{2}\sigma _{B}^{2}\geq \left[\max _{p_{k},\varrho _{k}}\sum _{k}p_{k}L(\varrho _{k})\right]^{2},$ █████ on the █████-████ ████ there is a ███████ ████ over the ██████████████ of the ███████ ██████. The ████████ ████████ above is █████████ by ███ ██████-█████ ███████ ██████.

With ███████ █████████, ███ can ██████ a ████████ with a ██████ ████ on the █████-████ ████ $\sigma _{A}^{2}F_{Q}[\varrho ,B]\geq 4\left[\min _{p_{k},\Psi _{k}}\sum _{k}p_{k}L(\vert \Psi _{k}\rangle \langle \Psi _{k}\vert )\right]^{2}$ █████ $F_{Q}[\varrho ,B]$ ███████ the ███████ ██████ ███████████ and the ███████ ██████ is ██████████ to ████ ██████ as $\varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .$ The ██████████ █████ █████████ of the ████ that the ███████ ██████ ███████████ is the ██████ ████ of the ████████ █████ ████.

A ███████ ██████████ ███████ without a ██████ ████ $\sigma _{A}^{2}F_{Q}[\varrho ,B]\geq \vert \langle i[A,B]\rangle \vert ^{2},$ which is ████████ than the ██████████ ███████████ ████████, since for the ███████ ██████ ███████████ ██ have $F_{Q}[\varrho ,B]\leq 4\sigma _{B},$ █████ for ████ ██████ the ████████ █████.

The ███████–████ ███████████ █████████

The █████████–███████████ ███████████ ████████ can be ███████ if the █████ of the ██████ is ██████ to be ██████████ of ███ of the ██████████. The ████████ ███████████ █████████ ██████ by ███████ ███████ and ████ █. ████ ████ ███-███████ ██████ on the ███ of the █████████ for ███ ████████████ ███████████. (███████ █████ on ███████████ █████████ ██████████ as the ███ of █████████ ███████, █.█., ███. ███ to ██████ █████.) For ███ ███-█████████ ███████████ $A$ and $█$ the █████ ████████ ███████████ ████████ is █████ by $\sigma _{A}^{2}+\sigma _{B}^{2}\geq \pm i\langle \Psi \mid [A,B]|\Psi \rangle +\mid \langle \Psi \mid (A\pm iB)\mid {\bar {\Psi }}\rangle |^{2},$ █████ $\sigma _{A}^{2}=\langle \Psi |A^{2}|\Psi \rangle -\langle \Psi \mid A\mid \Psi \rangle ^{2}$ , $\sigma _{B}^{2}=\langle \Psi |B^{2}|\Psi \rangle -\langle \Psi \mid B\mid \Psi \rangle ^{2}$ , $|{\bar {\Psi }}\rangle$ is a ██████████ ██████ that is ██████████ to the █████ of the ██████ $|\Psi \rangle$ and ███ ██████ ██████ the ████ of $\pm i\langle \Psi \mid [A,B]\mid \Psi \rangle$ to ████ this ████ ████████ a ████████ ██████.

The ██████ ████████ ███████████ ████████ is █████ by $\sigma _{A}^{2}+\sigma _{B}^{2}\geq {\frac {1}{2}}|\langle {\bar {\Psi }}_{A+B}\mid (A+B)\mid \Psi \rangle |^{2}$ █████ $|{\bar {\Psi }}_{A+B}\rangle$ is a █████ ██████████ to $|\Psi \rangle$ . The ████ of $|{\bar {\Psi }}_{A+B}\rangle$ ███████ that the █████-████ ████ of the ███ ███████████ ████████ is ███████ ██████ $|\Psi \rangle$ is an ██████████ of $(A+B)$ . ███ ███ ████ that $|\Psi \rangle$ can be an ██████████ of $(A+B)$ without being an ██████████ of ██████ $A$ or $█$ . ███████, ████ $|\Psi \rangle$ is an ██████████ of ███ of the ███ ███████████ the ██████████–███████████ ███████████ ████████ ███████ ███████. But the █████ █████ in the ███ ████████ is ███████ ██████ $|\Psi \rangle$ is an ██████████ of ████.

██████–████

An ██████–████ ███████████ ████████ like $\Delta E\Delta t\gtrsim \hbar /2,$ has a ████, █████████████ ███████; the ███████ of $\Delta t$ and $\Delta E$ ██████ and █████████ ████████████ have █████████ ██████ of ████████. ███████, ███ ████-█████ ███████████ is ████ ████ ███████████ and ██████████████ ████████: the ██████████ between the ████-████ of a █████████ █████, $\tau _{\sqrt {1/2}}$ and ███ ██████ █████ $\Delta E$ : $\tau _{\sqrt {1/2}}\Delta E=\pi \hbar /4.$ In ████████-███████, ██████ from ████████████ ████ to the █████–██████ ██████ ████████████ are used to ████████████ the ████████ of █████-██████ or ████████ ██████.