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Suppose the particles are initially in the (entangled) state $$A\otimes B+C\otimes D$$ where $A$ and $C$ are position eigenstates for particle 1 and $B$ and $D$ are position eigenstates for particle 2. Note that this state is the same as $$X\otimes Y+Z\otimes W$$ where $X=(1/2)(A+C)$ and $Y=(1/2)(A-C)$ are momentum eigenstates for particle 1 and ...

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I think the key point that you're missing is that as soon as you make a measurement, the entanglement between the two particles is broken. It should also be noted that the original particle also obeyed the uncertainty principle, and that at a quantum level there is no direct relationship between position, momentum and time. Another confusing factor is that ...

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Ok. Let's suppose that the initial state of the two particle are an eigenstate of the momentum operator (momentum is well defined). Quantum mechanics tell us that the position of the center of mass is not well defined. If we measure the position of the particle 1 (electron), then we do two thing in the system: We apply a measurement in a part of the whole ...

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Since you're an electronics student, I'll speak your language. Think of momentum and position as parameters in time and frequency domain of a signal rather than classical observable that are well defined. If you do so, you can easily realize that your frequency isn't well defined if you don't do an infinitely long measurement. This is simply due to the wave ...

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The problem is that you propose to make the two measurements "at time $t$ simultaneously". Measuring the particle's momentum cannot be done instantaneously; the more precisely you want to measure it, the longer the minimum required observation time becomes. (Rougly speaking that's because knowing the particle's momentum is equivalent to measuring its ...

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What is wrong with this logic is that you are supposing a particle has simultaneously well-defined position and momentum. This is not true - a state localized in real space is delocalized in momentum space, and vice versa. The classical conservation laws hold on the quantum level as operator laws, not as laws on the states.

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