$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Overview
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Two operators $A$ and $B$ are said to commute if $[A, B]=0$, where $[A, B]$ is a commutator defined as $$\begin{align*} [A, B]=AB-BA \end{align*}$$

Notations

The notation $[A, B]$ is shorthand used to denote the commutator, where $[A, B]=AB-BA$. If two operators commute, then we simply write $[A, B]=0$.

Concepts

If $A$ and $B$ commute, they have the same eigenstates

There's a proof for this
Let $A, B$ be two operators. If $A$ and $B$ commute, then $AB=BA$. If $A$ and $B$ don't commute, and $[A, B]=C$, then we can write $AB-BA=C$, or rearrange it into $AB=C+BA$ or $BA=AB-C$
If two operators commute, then they can be both measured at the same time. Otherwise, they cannot both be known at the same time.
Two operators $A$ and $B$ are said to commute if $[A, B]=0$, where $[A, B]$ is a commutator defined as $$\begin{align*} [A, B]=AB-BA \end{align*}$$

Notations

The notation $[A, B]$ is shorthand used to denote the commutator, where $[A, B]=AB-BA$. If two operators commute, then we simply write $[A, B]=0$.

Concepts

If $A$ and $B$ commute, they have the same eigenstates

There's a proof for this
Let $A, B$ be two operators. If $A$ and $B$ commute, then $AB=BA$. If $A$ and $B$ don't commute, and $[A, B]=C$, then we can write $AB-BA=C$, or rearrange it into $AB=C+BA$ or $BA=AB-C$
If two operators commute, then they can be both measured at the same time. Otherwise, they cannot both be known at the same time.
FullPage
Overview
Notations
Concepts
FullPage
Overview
Notations
Concepts
Two operators $A$ and $B$ are said to commute if $[A, B]=0$, where $[A, B]$ is a commutator defined as $$\begin{align*} [A, B]=AB-BA \end{align*}$$

Notations

The notation $[A, B]$ is shorthand used to denote the commutator, where $[A, B]=AB-BA$. If two operators commute, then we simply write $[A, B]=0$.

Concepts

If $A$ and $B$ commute, they have the same eigenstates

There's a proof for this
Let $A, B$ be two operators. If $A$ and $B$ commute, then $AB=BA$. If $A$ and $B$ don't commute, and $[A, B]=C$, then we can write $AB-BA=C$, or rearrange it into $AB=C+BA$ or $BA=AB-C$
If two operators commute, then they can be both measured at the same time. Otherwise, they cannot both be known at the same time.
Two operators $A$ and $B$ are said to commute if $[A, B]=0$, where $[A, B]$ is a commutator defined as $$\begin{align*} [A, B]=AB-BA \end{align*}$$

Notations

The notation $[A, B]$ is shorthand used to denote the commutator, where $[A, B]=AB-BA$. If two operators commute, then we simply write $[A, B]=0$.

Concepts

If $A$ and $B$ commute, they have the same eigenstates

There's a proof for this
Let $A, B$ be two operators. If $A$ and $B$ commute, then $AB=BA$. If $A$ and $B$ don't commute, and $[A, B]=C$, then we can write $AB-BA=C$, or rearrange it into $AB=C+BA$ or $BA=AB-C$
If two operators commute, then they can be both measured at the same time. Otherwise, they cannot both be known at the same time.
FullPage
Overview
Notations
Concepts