In quantum mechanics, observables are represented by Hermitian operators. Mathematically, two operators $\hat A$ and $\hat B$ commute if $$\tag 1 [ \hat A, \hat B]=\hat A \hat B- \hat B \hat A = 0$$

Commutators and commutation relations have a deep physical meaning in quantum mechanics. Hermitian operators that satisfy the commutation condition (1) have simultaneous eigenvalues, for example $a$, which are the observables, and eigenvectors, for example $\psi$ such that $$\tag 2 \hat A \psi = a \psi$$

Hermitian operators which satisfy (1) are also called compatible observables meaning that both can be measured simultaneously. This is of physical significance.

Hermitian operators which do not commute and do not satisy (1) are called incompatible observables, and a good example of this in quantum mechanics are the canonical commutation relations.

And operators or observables that commute with the Hamiltonian of the system are conserved quantities, e.g. angular momentum or spin. This means that these quantities to not change with time.

Commutation relations also do not depend on our choice of basis meaning they work just as well in coordinate space as they do in momentum space. This is a powerful result used in quantum mechanics.

In quantum mechanics, observables are represented by Hermitian operators. Mathematically, two operators $\hat A$ and $\hat B$ commute if $$\tag 1 [ \hat A, \hat B]=\hat A \hat B- \hat B \hat A = 0$$

Hermitian operators which satisfy (1) are also called compatible observables meaning that both can be measured simultaneously.

Hermitian operators which do not commute and do not satisy (1) are called incompatible observables, and a good example of this in quantum mechanics are the canonical commutation relations.

Now:

Operators or observables that commute with the Hamiltonian of the system are conserved quantities, e.g. angular momentum or spin. This means that these quantities do not change with time. Those that do not commute with the Hamiltonian, are not conserved quantities.

To summarise, if $\hat H$ is the Hamiltonian of the system then if

$$[\hat H, \hat O] = 0$$ then $\hat O$ is conserved and $$\frac{d \hat O}{dt}=0$$ but if $$[\hat H, \hat O] \ne 0$$ then $\hat O$ is not conserved and $$\frac{d \hat O}{dt} \ne 0$$

Commutation relations also do not depend on our choice of basis meaning they work just as well in coordinate space as they do in momentum space. This is a powerful result used in quantum mechanics.

In quantum mechanics, observables are represented by Hermitian operators. Mathematically, two operators $\hat A$ and $\hat B$ commute if $$\tag 1 [ \hat A, \hat B]=0$$

Commutators and commutation relations have a deep physical meaning in quantum mechanics. Hermitian operators that satisfy the commutation condition (1) have simultaneous eigenvalues $a$, which are the observables, and eigenvectors $\psi$ such that $$\tag 2 \hat A \psi = a \psi$$

Hermitian operators which satisfy (1) are also called compatible observables meaning that both can be measured simultaneously.

In quantum mechanics, observables are represented by Hermitian operators. Mathematically, two operators $\hat A$ and $\hat B$ commute if $$\tag 1 [ \hat A, \hat B]=\hat A \hat B- \hat B \hat A = 0$$

Commutators and commutation relations have a deep physical meaning in quantum mechanics. Hermitian operators that satisfy the commutation condition (1) have simultaneous eigenvalues, for example $a$, which are the observables, and eigenvectors, for example $\psi$ such that $$\tag 2 \hat A \psi = a \psi$$

Hermitian operators which satisfy (1) are also called compatible observables meaning that both can be measured simultaneously. This is of physical significance.

Hermitian operators which do not commute and do not satisy (1) are called incompatible observables, and a good example of this in quantum mechanics are the canonical commutation relations.

And operators or observables that commute with the Hamiltonian of the system are conserved quantities, e.g. angular momentum or spin. This means that these quantities to not change with time.

Commutation relations also do not depend on our choice of basis meaning they work just as well in coordinate space as they do in momentum space. This is a powerful result used in quantum mechanics.