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elemental09
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I'm still learning the formalism and structure of standard QM, so bear with me. My current confusion arises when considering the subsequent measurement of two commuting observables, for example the x-position and y-position observables of a single particle. Suppose the system is in a pure state [PLAIN][URL]http://latex.codecogs.com/gif.latex?\mid[/URL]&space;\Psi&space;\rangle [Broken].[/URL] Consider two commuting observables, [URL]http://latex.codecogs.com/gif.latex?\hat{A}[/URL] and [URL]http://latex.codecogs.com/gif.latex?\hat{B}.[/URL] Since these commute, there exists a basis of simultaneous eigenkets [URL]http://latex.codecogs.com/gif.latex?\mid[/URL] a,b\rangle [Broken], with [URL]http://latex.codecogs.com/gif.latex?\hat{A}\mid[/URL] a,b\rangle = a\mid a,b\rangle and [URL]http://latex.codecogs.com/gif.latex?\hat{B}\mid[/URL] a,b\rangle = b\mid a,b\rangle.
Now, a measurement of observable A on the system must result in an outcome of an eigenvalue, say [URL]http://latex.codecogs.com/gif.latex?a_{0}[/URL] of A. Further, the system immediately after the measurement is collapsed into the corresponding eigenstate [URL]http://latex.codecogs.com/gif.latex?\mid[/URL] a_{0},b_{0} \rangle [Broken], where [URL]http://latex.codecogs.com/gif.latex?b_{0}[/URL] is the eigenvalue of B corresponding to the eigenket the system was just collapsed into.
My question is: does this not mean that if one were to then measure observable B, the outcome would have to be [URL]http://latex.codecogs.com/gif.latex?b_{0}[/URL]? But how can this be true? After all, the two observables are compatible, and measurement of one should not affect the other. The two quantities, x-position and y-position, are independant even in QM, are they not? In other words, I should be able to measure the x-position of a particle andd obtain a definite result without altering the probability distribution of the outcomes of a subsequent y-position measurement.
Where does my confusion lie?
Now, a measurement of observable A on the system must result in an outcome of an eigenvalue, say [URL]http://latex.codecogs.com/gif.latex?a_{0}[/URL] of A. Further, the system immediately after the measurement is collapsed into the corresponding eigenstate [URL]http://latex.codecogs.com/gif.latex?\mid[/URL] a_{0},b_{0} \rangle [Broken], where [URL]http://latex.codecogs.com/gif.latex?b_{0}[/URL] is the eigenvalue of B corresponding to the eigenket the system was just collapsed into.
My question is: does this not mean that if one were to then measure observable B, the outcome would have to be [URL]http://latex.codecogs.com/gif.latex?b_{0}[/URL]? But how can this be true? After all, the two observables are compatible, and measurement of one should not affect the other. The two quantities, x-position and y-position, are independant even in QM, are they not? In other words, I should be able to measure the x-position of a particle andd obtain a definite result without altering the probability distribution of the outcomes of a subsequent y-position measurement.
Where does my confusion lie?
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