I have heard if we want to obtain classical results from quantum mechanics, we have to choose the commutator of $\hat X$ and $\hat P$ to be $\left[\hat X,\hat P\right]=i\hbar$. Is there any reason support this statement?
EDIT: I want to understand how inventors of QM deduced the $\hat X$ and $\hat P$ should not commute. I really wonder why numerous lecturers postulate it.
The idea goes back to Heisenberg. He believed that physics could only describe quantities that could be measured experimentally, and sought to develop a mathematical theory that would reflect this and correctly predict the relative intensities of spectral lines.
In classical physics, the radiated intensities depend (in the first approximation) on electric dipoles, which are dependent on the position of the electrons. To account for the fact that the position of the electrons cannot be measured during a transition, he introduced a number $x_{nm}$ to characterize the position in the transition from $n\to m$. He also introduced $v_{nm}$ for the velocities of the electrons during the transitions and a related acceleration $a_{nm}$.
Heisenberg was eventually able to reproduce the energy levels $E_n$ (actually their differences $E_n-E_m$) using these quantities but only if the quantities satisfied the "unusual" combination properties $$ \sum_{m} x_{nm} v_{mk} = A_{nk} \ne A_{kn} = \sum_m v_{km} x_{mn}\, . $$ In particular, using his "tables" of $x_{nm}$ and $v_{nm}$ he was able to work out what we now write as $[x,p]$.
The story goes that Pascual Jordan met Heisenberg in a train at the time Heisenberg was working on this. Jordan, who had mathematical training, recognized the combination rule as matrix multiplication. Jordan, together with Max Born and Paul Dirac, realized that the use of non-commutative quantities was essential to Heisenberg's description.
Dirac in particular postulated that the multiplication rules had to follow from dynamical considerations; inspired by the correspondence principle, he was able to relate, up to an overall factor, the classical Poisson bracket to a quantum bracket to find the now-famous $$ [q_i,q_j]=0\, ,\qquad [p_i,p_j]=0\, ,\qquad [q_j,p_k]=i\hbar \delta_{jk}\, . $$
There are several accounts of this discovery. The most historical is by Max Jammer, who was able to interview firsthand some of actors in the story. There is also an interesting and more recent text by Roland Omnes but it doesn't focus so much on the history. I'm sure there are others.
Edit: After reading @hyportnex account, I found Jammer online and checked. hyportnex account is accurate when it comes to Born recognizing the matrix form of Heisenberg's expression. As to the story of the train: it is Born that met Jordan in a train. Quoting from Jammer, page 109:
Now it happened that Born, while traveling by train to Hanover, told a colleague of his from Gottingen about the fast progress of his work but also mentioned the peculiar difficulties involved in the calculations with matrices. It was fortunate and almost an act of providence that Jordan, who shared the same compartment in the train, overheard thispiece of conversation. At the station in Hanover Jordan then introduced himself to Born, told him of his experience in handling matrices, and expressed his readiness to assist Born in his work.
To study the history of the subject I recommend van der Waerden: Sources of Quantum Mechanics (Dover Publications). This book only discusses the history of matrix mechanics, so the development of wave mechanics (de Broglie, Schrodinger, etc.) is left for a never finished/published 2nd volume. To write this book Waerden contacted the main actors directly: Pauli, Heisenberg, Born, Jordan, etc. Let me quote a letter that Waerden received from Born, see pages 36-37.
Born's conjecture on pq — qp
On July 19, Born took the train to Hannover to attend the meeting of the Deutsche Physikalische Gesellschaft. His own account, confirmed by Jordan's testimony, runs thus:
After having sent Heisenberg's paper to the Zeitschrift fur Physik for publication, I began to ponder about his symbolic multiplication, and was soon so involved in it that I thought the whole day and could hardly sleep at night. For I felt there was something fundamental behind it ... And one morning ... I suddenly saw light: Heisenberg's symbolic multiplication was nothing but the matrix calculus, well known to me since my student days from the lectures of Rosanes in Breslau. I found this by just simplifying the notation a little: instead of $q(n, n+\tau)$ I wrote $q(n, m)$, and re-writing Heisenberg's form of Bohr's quantum conditions I recognised at once its formal significance. It meant that the two matrix products $\mathbf{pq}$ and $\mathbf{qp}$ are not identical. I was familiar with the fact that matrix multiplication is not commutative; therefore I was not too much puzzled by this result. Closer inspection showed that Heisenberg's formula gave only the value of the diagonal elements (m=n) of the matrix pq — qp: it said that they were all equal and had the value $h/i2\pi$. But what were the other elements $m \ne n$ ? Here my own constructive work began. Repeating Heisenberg's calculation in matrix notation, I soon convinced myself that the only reasonable value of the non-diagonal elements should be zero, and I wrote down the strange equation
$$\mathbf {pq-qp} = \frac{h}{2\pi i} \mathbf{I}$$ where $\mathbf{I}$ is the unit matrix. But this was only a guess, and my attempts to prove it failed.
The commutator arises in canonical quantisation which is a procedure to quantise a classical theory, ubiquitous in most quantum mechanics and quantum field theory texts. One imposes the condition that,
$$[x,p] = i\hbar$$
which is in accordance with the rule of thumb (as there are many subtleties as to its applicability),
$$\{x,p\} \to \frac{1}{i\hbar}[x,p]$$
attributed to Dirac. The quantisation is a means of going from a theory specified by either an action or a phase space with a particular symplectic structure. So, your question boils down in essence to the validity of quantisation in this manner.
In order to rigorously justify this, I would recommending reading the nLab article; it has a habit of over-complicating certain matters especially with jargon but I think will address your concerns.
Another way to see it is by trying to define $x$ and $p$ in a quantum theory sensibly. For example, the momentum operator has an intuitive meaning and an expected behaviour on states.
Furthermore, in for example quantum field theory, one can apply Noether's theorem in the case of translational symmetry to derive the conjugate momentum, and then upon quantising the fields of the theory, modulo potential ordering ambiguities, one arrives at an operator $p$.
Explicitly, one may write out $\phi(x)$ and $\pi(x)$ say, and check that,
$$[\phi(x),\pi(x)] = i\hbar \delta^{(3)}(x-y)$$
which is the analogue of $[x_i,p_j] = i\hbar \delta_{ij}$ though this derivation would depend on the fact that $[a,a^\dagger]= 1$ for the ladder operators and so the two commutators' definitions are intertwined in a sense. Asking for a justification of the canonical relations is linked to the relation for ladder operators.
