Popper's Experiment - Criticism of Popper's Proposal

Criticism of Popper's Proposal

Tabish Qureshi has published the following analysis of Popper's argument.

The ideal EPR state is written as, where the two labels in the "ket" state represent the positions or momenta of the two particle. This implies perfect correlation, meaning, detecting particle 1 at position will also lead to particle 2 being detected at . If particle 1 is measured to have a momentum, particle 2 will be detected to have a momentum . The particles in this state have infinite momentum spread, and are infinitely delocalized. However, in the real world, correlations are always imperfect. Consider the following entangled state

$\psi(y_1,y_2) = A\!\int_{-\infty}^\infty dp e^{-p^2/4\sigma^2}e^{-ipy_2/\hbar} e^{i py_1/\hbar} \exp$

where represents a finite momentum spread, and is a measure of the position spread of the particles. The uncertainties in position and momentum, for the two particles can be written as

The action of a narrow slit on particle 1 can be thought of as reducing it to a narrow Gaussian state: . This will reduce the state of particle 2 to . The momentum uncertainty of particle 2 can now be calculated, and is given by

$\Delta p_{2} = \sqrt{\frac{\sigma^2(1+\epsilon^2/\Omega^2)+ \hbar^2/16\Omega^2}{1+4\epsilon^2(\sigma^2/\hbar^2+1/16\Omega^2)}}.$

If we go to the extreme limit of slit A being infinitesimally narrow, the momentum uncertainty of particle 2 is, which is exactly what the momentum spread was to begin with. In fact, one can show that the momentum spread of particle 2, conditioned on particle 1 going through slit A, is always less than or equal to (the initial spread), for any value of, and . Thus, particle 2 does not acquire any extra momentum spread than what it already had. This is the prediction of standard quantum mechanics. So, the momentum spread of particle 2 will always be smaller than what was contained in the original beam. This is what was actually seen in the experiment of Kim and Shih. Popper's proposed experiment, if carried out in this way, is incapable of testing the Copenhagen interpretation of quantum mechanics.

On the other hand, if slit A is gradually narrowed, the momentum spread of particle 2 (conditioned on the detection of particle 1 behind slit A) will show a gradual increase (never beyond the initial spread, of course). This is what quantum mechanics predicts. Popper had said

...if the Copenhagen interpretation is correct, then any increase in the precision in the measurement of our mere knowledge of the particles going through slit B should increase their scatter.

This particular aspect can be experimentally tested.

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