The theory that is given to us by Nature is not classical mechanics, but quantum mechanics. Any attempt to ``derive'' quantum mechanics from classical will eventually fail due to ambiguities. There are analogies, of course. But rather than viewing them as a basis for deriving QM from classical, we should rather view these relations as arising from the fact that, in the limit in which the actions characteristic of the system under consideration are much larger than hbar, the predictions of quantum mechanics must become identical to those of classical mechanics.

QM can, for example, be rewritten exactly in a way that involves some kind of phase space. See for example:

https://en.wikipedia.org/wiki/Phase_space_formulation

There, a non-commutative star product is used, for example. Heisenberg's equation then looks like

dot A = i(H*A - A*H)/hbar

(or maybe the other way around). In the limit hbar->0, the expression explicitly tends to the Poisson bracket.

There are other ways to understand the semiclassical limit (Feynman path integrals for instance). But in all cases, seeing the classical structure emerge from the *given* quantum theory in the hbar->0 limit is what explains, in my view, the connections between the two, not viceversa.

Of course, when physicists did not yet really know what the form of QM was going to be, they let themselves be guided by analogies. This is just fine. But now that we know QM, we should not view these analogies as having any essential significance.

The classic book Goldstein talks about the Poisson bracket and canonical variables in detail but it does not link that to commutation relation to bridge with quantum mechanics. Rather it transits to quantum fields starting from classical fields in the last chapter. Still it is better to go through this book first as its linking is a short idea you can find in other classical mechanics book if you skim through few of them. 

Hi Swati! To answer this interesting question you need to remind each other stories of quantum physics, which began in 1900 by Planck discovery, later Einstein photoelectrical description in 1905, later almost simultaneously finding equations of QM by Schrodinger and Heisenberg in 1925-6, and yet later Dirac's works about the equivalence of the two formulations, and apart of this it would be necessary to answer what they led to these discoveries, And perhaps it would be enough to answer your question:-)? But remember that  geniuses like Bohr, Richard Feynman and others did not understand all:-(

Jurek H

The theory that is given to us by Nature is not classical mechanics, but quantum mechanics. Any attempt to ``derive'' quantum mechanics from classical will eventually fail due to ambiguities. There are analogies, of course. But rather than viewing them as a basis for deriving QM from classical, we should rather view these relations as arising from the fact that, in the limit in which the actions characteristic of the system under consideration are much larger than hbar, the predictions of quantum mechanics must become identical to those of classical mechanics.

QM can, for example, be rewritten exactly in a way that involves some kind of phase space. See for example:

https://en.wikipedia.org/wiki/Phase_space_formulation

There, a non-commutative star product is used, for example. Heisenberg's equation then looks like

dot A = i(H*A - A*H)/hbar

(or maybe the other way around). In the limit hbar->0, the expression explicitly tends to the Poisson bracket.

There are other ways to understand the semiclassical limit (Feynman path integrals for instance). But in all cases, seeing the classical structure emerge from the *given* quantum theory in the hbar->0 limit is what explains, in my view, the connections between the two, not viceversa.

Of course, when physicists did not yet really know what the form of QM was going to be, they let themselves be guided by analogies. This is just fine. But now that we know QM, we should not view these analogies as having any essential significance.

For one thing, the Poisson bracket for functions F(q,p) of the canonical coordinates obeys the same basic algebraic structure as the commutator of the corresponding operators for q, p.

We can then notice that there is a fundamental Poisson bracket

{q,p}=1

Dirac pointed out that this is a very natural point to introduce Quantum Mechanics, by insisting that, in addition to replacing q, p with non commuting operators, one defines their commutator [ , ] via the replacement

{ , } -> [ , ]/(i \hbar)

The fundamental Poisson bracket then becomes

[\hat q,\hat p]=i\hbar

and we recover one of the basic rules of quantum mechanics.