# Relationship between symmetries and quantum operators of classical quantities?

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?

• 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 $$\hat{Q}\propto\frac{\partial}{\partial q}$$ in $$q$$-space.