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How do you think about the uncertainty principle for two particles that are interacting with each other?

As you know well, the Heisenberg's uncertainty principle is well known by the inequality like,

(delta x)(delta p)>h/2         (1)

for a single particle.  Then, what will the inequality be like for a two-body system?  Suppose that two particles A and B are flying in space and are interacting with each other.  And assume that there is a detector screen which is placed vertically on the right side to the two particles.  Some time, before long, one of the two particles will collide with the detector screen, and on that instance its positional uncvertainty will vanish.  That is, its wave nature vanishes or collapses and particle nature appear as a dot on the detector screen instead and suddenly.  At such an instance, what do you think is happening to the second particle?  Does the detection of the first particle affect the state of the second particle?  Does it affect the uncertainty of the second particle?  If that were to be the case, then how?  What will the uncertainty inequality for the second particle be like?

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