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So when you have an observable which is the measurement operator acting on the state you get a different result than in a different basis. What does it mean in physical terms? Does that mean writing your space operation in different style could collapse your state to something different?
Again, we can understand it mathematically but what's in physical terms?
One way to think about it is that "measuring in a given basis" is how we describe mathematically the act of interacting with the system in different ways.
Taking as an example a qubit, "measuring in the $\{\lvert0\rangle,\lvert1\rangle\}$ basis" is mathematese for "interacting with the system in a way that reveals to the experimenter whether the system is in the physical state which we are thinking of as corresponding to $\lvert0\rangle$ or $\lvert1\rangle$". In general, the system will not be in a state that can be directly associated with one of these basis states, in which case the act of measurement "forces" the system to collapse to one of the two states, and this happens with a probability that depends on the actual state of the system, and induces a physical change in the state of the system. Measuring in a different basis then simply corresponds to another type of physical interaction with the system.
Another, slightly more informal way to think about it is to say that each measurement basis corresponds to a different "question" being asked to the system. "Measuring in $\{\lvert0\rangle,\lvert1\rangle\}$" means to be asking the system whether it's in the $\lvert0\rangle$ or $\lvert1\rangle$ state, while "measuring in $\{\lvert+\rangle,\lvert-\rangle\}$" means to be asking the system whether it is in the $\lvert+\rangle$ or $\lvert-\rangle$ state. As it happens, in quantum mechanics the act of "asking a question" to the system - that is, of collecting some classical information out of the system - will in general result in a perturbation of the system itself, which in quantum mechanics is captured by means of the noncommutativity of different observables.
So when you have an observable which is the measurement operator acting on the state you get a different result than in a different basis. What does it mean in physical terms?
@glS has already explained the physics. I'll give an example of an observable: photon polarization. You'll find an elaborate description here. Measurements in this context would simply mean passing the photons through linear polarizers (in various orientations). If you can write the out the quantum state, using any appropriate orthogonal basis (one basis vector would be in the direction of the slit of linear polarizer and another would be in a direction orthogonal to it) you can predict the percentage of photons that will come out of the polarizers. You can compare the results with Malus' law.
Does that mean writing your space operation in different style could collapse your state to something different?
Not exactly sure what you mean by "space operation" (measurement is not an operator!), but I think you're on the right track. If you measure the observable in a certain basis (say, pass any arbitrary linearly polarized light through a horizontal-vertical dual polarizer), the photons which will pass through the polarizer will either be horizontally polarized or vertically polarized. Similarly, if you pass any arbitrary linearly polarized light through a diagonal-antidiagonal dual polarizer the photons which will pass through will either be diagonally polarized or anti-diagonally polarized. Does this help with the intuition?
