# Link between Quantum and Classical Mechanics [duplicate]

In classical mechanics we have momentum as generator of translation by following definition:

$$f(x+\delta x)=f(x)+[f(x),p]\delta x+....$$

I was wondering whether using this relation and commutation relation between $\hat{x}$ and $\hat{p}$ can we come to a quantum mechanic relation of momentum as generators:

$$\left(1-\frac{i}{\hbar}\hat{p}dx\right)|x\rangle=|x+dx\rangle$$

I am preferring to use commutation relationships as they are the bridge between quantum mechanics and classical mechanics.

• I got the answer....The first equation indeed find its place in Quantum mechanics for observables in Heisenberg's picture where also it treats momentum as generator of translation..this what my concern was. Aug 3 '14 at 13:39