15.2: What does it mean - Wavefunction collapse and the EPR paradox

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The interpretation of collapsing wavefunctions is often regarded as unphysical, or philosophically problematic. There appears to be a contradiction with relativity in the idea that the wavefunction collapses instantaneously throughout space, although the wavefunction is not measurable.

An attractive contrary view to the idea of ‘measurement collapsing the wavefunction’ is that for a particular system the value of a observable is a property of the particle, and the wavefunction only expresses averages over many particles. This kind of property is known as a hidden variable. As we shall see, this interpretation of quantum mechanics can be tested, and is inconsistent with experimental results.

Consider a two-photon decay from a source (e.g. \(^{40}\)Ca). Two polarisers are oriented along the z-direction, and we detect whether or not the photons pass through the polariser.

The decay is one in which angular momentum is conserved, so the photons must be either both right-polarised \(({\bf e}_R)\) or both left-polarised \(({\bf e}_L)\) (they travel in opposite directions). We are dealing with bosons, so the wavefunction can be written as a superposition:

\[|12 \rangle = \sqrt{\frac{1}{2}} ({\bf e}_{1R}{\bf e}_{2R} + {\bf e}_{1L} {\bf e}_{2L}) \nonumber\]

Now convert into x and y polarization using \({\bf e}_R = ({\bf e}_x − i{\bf e}_y)\) and \({\bf e}_L = ({\bf e}_x + i{\bf e}_y)\) to give

\[|12 \rangle = \sqrt{\frac{1}{2}} ({\bf e}_{1x}{\bf e}_{2x} + {\bf e}_{1y}{\bf e}_{2y}) \nonumber\]

From this we can clearly see that the quantum probability of the photon 1 passing through its detector is \(\frac{1}{2}\), and if so the wavefunction collapses onto \(|12 \rangle = {\bf e}_{1x}{\bf e}_{2x}\) and the conditional probability of the second photon passing through its detector is then 1. Thus quantum mechanics tells us that the probability of both detectors counting is \(\frac{1}{2}\).

Contrariwise, a hidden variables argument might say that on production the photons were polarised in a random direction, say \(\theta\) to the x-axis. In this case the probability of passing through either detector would be \(\cos^2 \theta\), and the probability of simultaneous counts will be \(\langle \cos^4 \theta \rangle = 3/8\). The mathematics for particles with correlated spins is similar.

Since the wavefunction collapse and hidden variable approach give different answers, we can do an experiment to see which is correct.