Relationship between symmetries and quantum operators of classical quantities?

Question

I noticed this the other day. I don't really know "what" this means, I'd love to understand.

The energy operator is $hat E = -i hbar frac{partial}{partial t}$. Conservation of energy is a consequence of time symmetry.
The momentum operator is $i hbar frac{partial}{partial x}$.  Conservation of momentum is a consequence of space symmetry.
The angular momentum operator is $-i hbar (r times nabla)$. Conservation of angular momentum is a consequence of rotational symmetry, which 'feels related' to curl: $r times nabla$.

Is the "general form" of any quantum mechanical operator of a given classical quantity $Q$, whose conservation law is given by a symmetry in some 'direction '$d$ going to be proportial to $hat Q equiv i hbar frac{partial}{partial d}$?
If  not, why do the energy and momentum operator have their symmetries in the derivative? is there a reason?

Qmechanic · Accepted Answer

There is a lot of truth behind OP's observations, which are backed up by the following facts:

An infinitesimal symmetry $delta$ with symmetry parameter $epsilon$ is generated by a Noether charge $hat{Q}$ in the sense that $delta=epsilon [hat{Q},cdot]$, cf. e.g. this Phys.SE post.

The symmetry parameter $epsilon$ can often be associated with a variable/coordinate $q$ of theory.

If $[hat{Q},q]propto {bf 1}$ is proportional to the identity operator we can go to the corresponding Schroedinger representation
$hat{Q}proptofrac{partial}{partial q}$ in $q$-space.

Relationship between symmetries and quantum operators of classical quantities?

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