What is the physical significance of commutation between position and momentum operator?

Answer 1

Ah, a philosophical and mathematical question.

Mathematically speaking, it means that you cannot operate on a wavefunction with either the position or momentum operator and then the other without getting different answers. When you operate on a wavefunction, you change it mathematically. Depending on the order of operation, you will get different answers.

Philosophically, this starts to bring up an odd idea that you can know one thing about a system, but not the other simultaneously. his is where the Heisenberg Principal of Uncertainty comes into play. Classical physics disagrees with this statement, but so far we are forced to accept it.

Physically, this means that when you make a physical, actual real life observation of a system, you change or affect it. (This is part of the Copenhagen interpretation of quantum mechanics, and there are physicists who disagree with it.) Nonetheless, this is what the mathematics tells us. When you look at a particle and ask it "where are you", the particle "collapses" to a certain state, and you can no longer know it's momentum. The opposite is true as well.

Answer2:

Excellent background above of the issue of the philosophical question of commutation. The mathematical and physical answer is this, position R and momentum P are vectors. Vectors non-commute, i.e. RxP = - PxR. This means physically if RxP points up, then PxR will point down. If RxP is spin then spin is reversed with PxR, thus RxP - PxR = 2 RxP and RP - PR = 2RxP is also true because R and P are Quaternions, the sum of scalar and 3 vectors, P=[p,P],

[p,P][r,R] - [r,R][p,P]=[ pr - P.R, pR + Pr + PxR] - [rp - R.P, rP + Rp + RxP]=2[PxR]

With regard to "uncertainty", IF" h=rp" and 'h' is known with "certainty" and one measures' r' with certainty, then p is known with certainty, as' h/r'=p !

Now if 'h or r' are not known with certainty, neither can p. I presume that 'h' Planck's Constant is known, then if 'r' or 'p' can be measured the other is known with physics certainty, e.g c speed of light.

The problem may lie in not recognizing the Quaternion Space underlying Quantum Theory. Dirac's Rules are isomorphic to Quaternion Rules. Unfortunately, Quaternions are not taught in math and Physics.