Commutator of x and p in quantum mechanics

[rbphysics]

TL;DR

Is it a postulate?

The commutator of [x,p]=i$\hbar$. Is it a postulate? No book state it as postulate of Quantum mechanics. But, I don't see anything more general by which I can derieve this. At elementary level Quantum mechanics, one start with momentum operator in position representation to derieve this. But that would mean to take momentum operator as postulate. But If we take the commutator as postulate one can derieve momentum operator in position representation as well as position operator in momentum repres

 

[gentzen]

But that would mean to take momentum operator as postulate.

Momentum is the Noether current associated with translation invariance. The momentum operator is the generator of translations. This is the principle from which the momentum operator follows.

But where does $\hbar$ occur in this? It fixes the units, but its numerical value needs to be measured, or „postulated“ (with respect to your question).

 

[anuttarasammyak]

Following your interest to find more profound idea in quantum mechanics, Heisenberg and Dirac observed that Poissson Bracket {A,B} which appears in analytic mechanics, especially in Hamilton's equation of motion ( https://en.wikipedia.org/wiki/Poisson_bracket ) has its quantum version
$$ \frac{AB-BA}{i\hbar}$$
With this replacement all the classical mechanics structure becomes available in QM also.
As an example in CM, $$\{x,p\}=1$$ thus in QM
$$ \frac{xp-px}{i\hbar}=1$$

 

[pines-demon]

TL;DR Summary: Is it a postulate?

The commutator of [x,p]=i$\hbar$. Is it a postulate? No book state it as postulate of Quantum mechanics. But, I don't see anything more general by which I can derive this. At elementary level Quantum mechanics, one start with momentum operator in position representation to derieve this. But that would mean to take momentum operator as postulate. But If we take the commutator as postulate one can derive momentum operator in position representation as well as position operator in momentum repres

A common way to quantize classical systems ("canonical quantization") is indeed to take conjugate variables $q,p$ that follow Poisson bracket relation $\{q,p\}=1$ and swap it by $[\hat q,\hat p]=i\hbar$. So in this formalism, it is indeed a postulate.

 

[andresB]

TL;DR Summary: Is it a postulate?

The commutator of [x,p]=i$\hbar$. Is it a postulate? No book state it as postulate of Quantum mechanics. But, I don't see anything more general by which I can derieve this. At elementary level Quantum mechanics, one start with momentum operator in position representation to derieve this. But that would mean to take momentum operator as postulate. But If we take the commutator as postulate one can derieve momentum operator in position representation as well as position operator in momentum repres

You can take it as a postulate, and together with the algebra of the space-time transformations you can derive Quantum mechanics. See for example the book by Ballentine.