If we have two classical quantities $A$, $B$, and their corresponding quantum operators $\hat{A}$, $\hat{B}$, then their commutators and Poisson brackets are linked by
$$ [ \hat{A},\hat{B} ] \rightarrow i \hbar \text{{A, B}}.$$
Why is this the case? I couldn't find a simple explanation.
Notice that the Poisson bracket and the commutator are both Lie brackets.
The fundamental CCRs are often taken$^1$ as a first principle in QM, while the fundamental Poisson brackets are a definition in classical mechanics. This establishes the $i\hbar$ normalization. (The factor $i$ is e.g. discussed in this Phys.SE post.)
The Poisson bracket satisfies Leibniz rule, while the commutator does not, so the correspondence is expected to be deformed for composed observables, cf. e.g. my Phys.SE answer here.
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$^1$ Most QM textbooks define the momentum operators $\hat{p}_i$ as (appropriately normalized) generators of position translations from which the CCRs follow.
The product A B always is a p-q-monomial, that has any dimension in powers of length and momentum with product dimension in classical phase space. Monomials represent obervable physcal quanties used to calculate classical expectation values as all moments in a given probability distribution with integrand
$$ A(p,q) \quad B(p, q) \quad (dp \ dq)^n =\left( \sum a_{i k} \ p^i \ q^k \quad b_{j l} \ p^j \ q^l \right) \quad (dp\ dq)^n $$
The phase space product $dp dq$ has dimension $$x * m x/t = m x^2/t = (m x^2/t^2)/ t = m x\times v *\phi $$ that is called action and pairs displacement and momentum, time displacement and energy or angular momentum and angle displacement.
The fundamental significance of the action of the paired product $dp^i\wedge dq^i$ is Liouvilles theorem: all volumes and expectations are constant in time along solutions of the canonical equations of motion (models independent on time explicitely).
With Plancks constant $h$ in the canonical ensemble of radiation in a box as the discrete energy level spacing in terms of the lowest wavelength
$$\lambda = L/2, \lambda \omega = c, E_n = n \hbar \omega $$
the classical conundrum of the action dimension could be removed on the classical level simply by rescaling the momentum in units of inverse length:
$$p = \hbar k = 2 \pi h /\lambda $$
By Liouville's theorem, the phase space product of the pq-differential is constant with respect to any canonical transformation including all coordinate transformations, if the momentum is transformed ba the inverse transformation.
Bottom line: All Lagrangrians and Hamiltonians in physics have to be expressed in terms of x[lenght] and p[1/lenght] with an invariant universal $\hbar/length, \hbar \partial_x $ in units of momentum, $\omega \hbar, \hbar \partial_t $ in units of energy and $\hbar$ as unit of angular momentum.
Only after detection of the quantum Hall effect by von Klitzing it became clear, that any $\hbar$ in physics is tied to the unit of elementary charge via the (in)famuos, dimensionless fine structure constant of electromagnetic interaction
$$ \alpha = \frac{e^2 }{2 h \ \varepsilon_0 \ c }= 1/136.9...$$
Now this number is one of the open questions in the foundations of elementary particle physics with an uncountable number of attempts to find an equation with this solution. No success until todays because its the number with the most known number of digits, experimentally confirmed.
Find the reason why electric charge is, lets say, about $\frac{1}{10}$
$$e= \frac{1}{\sqrt{\frac{136.\dots}{4 \pi}}} $$
Latest spectacular attempt by late Sir Michael Atiyah
