Generally in classical mechanics, the phase space of a measurable is spanned by the coordinate and momentum . In other words, we can describe any classical measurable by:

Naturally we could use chain rule, if we want to see the time-evolution of this measurable :

Switch to Hamiltonian mechanics

To look for a compact representation of our classical measurable, we study the Hamiltonian of any classical system. We know that the Hamiltonian consists of the kinetic energy and the potential energy as a whole:

Notice that,

1. Conservation of energy if potential energy is TI:
   
   

2. .

3. .

Put the results above into the EOM of , we obtain:

This motivates the introduction of a new nomenclature: Poisson Brackets. For measurable and in the phase space , let’s denote:

With this new notation, the EOM of now becomes:

And as a by-product, we discover as well:

For a quantity that is conserved, must be and thus must vanish. This is essentially the Hamiltonian version of Noether’s theorem.

From Classical to Quantum

Heisenberg EOM reads:

This EOM remains almost the same form, even when observables are turned into operators living in the Hilbert space!

Appendix

Classical and quantum canonical commutators: