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If I have 2 observable operators $A$ and $B$, if $A$ and $B$ commute: $[A, B] = 0$, then they must necessarily form a complete set of commuting observables (CSCO). Essentially, if 2 observables are compatible this seems to be quite significant.

I just wanted to get an intuition for what this means. Does it have something to do with precision of measurement?

For example, I know that if the Hamiltonian is time independent, it commutes with itself: $$[H, H] = 0$$ However, if the Hamiltonian is time-dependent, then this is not true at all times: $$[H(t_1), H(t_2)] \neq 0$$ Is this because the Hamiltonian is changing and thus doesn't necessarily act on itself the same way at all times anymore?

Also I know that the position and momentum operators commute if they are in different directions, but don't commute if they're in the same direction: $$[x_i, p_j] = \delta_{ij}$$ Does this imply that if 2 observables don't commute, this corresponds to the idea that we can't measure them both simultaneously to high precision?

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  • $\begingroup$ A set of CSCO per definition is such that all possible common eigenstates are uniquely determined by the eigenvalues of these observables. In terms of classical states, that the state is uniquely determined by the values of these observables. $\endgroup$ – doetoe Nov 9 '18 at 13:36
  • $\begingroup$ Related to conservation law if an observable commutes with the Hamilton. $\endgroup$ – K_inverse Nov 9 '18 at 13:43
  • $\begingroup$ @K_inverse Could you elaborate a little more on that? If for example $[S_x, H] = 0$, this implies some spin conservation law? $\endgroup$ – CuriousHegemon Nov 9 '18 at 13:48
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    $\begingroup$ @CuriousHegemon, FYI en.wikipedia.org/wiki/Heisenberg_picture $\endgroup$ – K_inverse Nov 9 '18 at 15:10
  • $\begingroup$ That's right, the Heisenberg Equation of motion! $\frac{dA}{dt} = \frac{1}{i \hbar} [A, H]$. So if $A$ commutes with $H$, then $\frac{dA}{dt} = 0$. Thanks! $\endgroup$ – CuriousHegemon Nov 9 '18 at 15:35
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In quantum mechanics, a measurement (almost) always modifies the system being measured. Intuitively, if two measurements commute then the way that one measurement changes the system doesn't affect the results of the other measurement. So if you repeat those two measurements as many times as you want, in any order, then you'll always get the same results for each of the two measurements.

On the other hand, if the two measurements don't commute, then every time you perform one, it will (at least partially) "reset" the other observable to an indeterminate value (I'm glossing over some subtleties).

This explains why if an observable commutes with the Hamiltonian, then its value is conserved over time: the Hamiltonian "pushes the system into the future", so in some sense it is continuously acting on the system and changing it in a way that modifies the value of most observables - except for the special ones that commute with the Hamiltonian, and so are not affected by time evolution.

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    $\begingroup$ Great answer, thank you. I think it makes a lot more sense now. So if we have two observables $A$ and $B$, then if $AB|\psi\rangle \neq BA|\psi\rangle$, where $|\psi\rangle$ is a state ket, this implies that it matters the order in which we measure A and B, implying the measurement of either A or B affects the measurement of the other. Awesome! $\endgroup$ – CuriousHegemon Nov 9 '18 at 15:37
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From Wikipedia, the free encyclopedia

In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.[1]

Since each pair of observables in the set commutes, the observables are all compatible so that the measurement of one observable has no effect on the result of measuring another observable in the set. It is therefore not necessary to specify the order in which the different observables are measured. Measurement of the complete set of observables constitutes a complete measurement, in the sense that it projects the quantum state of the system onto a unique and known vector in the basis defined by the set of operators. That is, to prepare the completely specified state, we have to take any state arbitrarily, and then perform a succession of measurements corresponding to all the observables in the set, until it becomes a uniquely specified vector in the Hilbert space.

[1] Gasiorowicz, Stephen (1974), Quantum Physics, New York: John Wiley & Sons, ISBN 978-0-471-29281-4.

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