After my post on Einstein, Podolski and Rosen’s paradox, a few of you guys asked me about how is it possible for both the position and momentum of a particle to be elements of reality at the same time. In particular, I think this confusion may come from the choice of example I used (Eq. (3)). I decided to prepare this addendum to present another example of how entangled particles can apparently allow us to determine the values of two non-commuting operators. Thus, I want to discuss Bohm’s version of the EPR paradox. (Also, this article will be very useful for understanding Bell’s Inequalities).

Before presenting Bohm’s experiment I want you to remember that EPR’s criterion of reality establishes that in order for a physical quantity to have real values we should be able to predict with certainty (i. e. with probability equal to unity) the value of said quantities without the need of measuring or disturbing the system.

Bohm’s Experiment

In the years that followed the EPR paper, their thought experiment continued to puzzle physicist, but it remained just that, a though experiment. It wasn’t until fifteen years after the paper was published that David Bohm proposed an experiment that could exemplify the paradox using the spins of a pair of particles.

Think of the following situation: a particle with zero spin decays into two particles (A and B), each with 1/2-spin. Due to the fact that spin angular momentum must be conserved during the decay, if initially the total spin angular momentum was zero, then after the decaying process it must still be zero. Therefore, particles A and B have opposite spin.

Take as an example the dissociation of an exited hydrogen molecule into two hydrogen atoms. If the decaying mechanism does not change total angular momentum, then the spins on the hydrogen atoms will be anti-correlated. Whenever a measurement of the spin of A is found to be positive with respect of the z-axis (we shall note this state as $|\!\uparrow\rangle$ ), then, we could infer that the spin of the B particle must be negative ( $|\!\downarrow\rangle$ ), and this is true independent of the distance that separates the particles. The spin of these particles are then entangled.

Now, we can´t predict which spin will be positive (or negative) with respect of the z-axis, so the state that describes the spins of the particles could be for instance the spin singlet state

$|\psi\rangle=\frac{|\downarrow\uparrow\rangle-|\uparrow\downarrow\rangle}{\sqrt{2}}$ . (1)

We have a probability of 50% for the spin of particle A to be positive (and the spin of B negative) and a probability of 50% of it being the other way around.

So far we have assumed that we are performing a measurement along the z-axis, but measurements are not restricted to this particular election, we could measure for instance the spin of particle A along the $\mathbf{\check{a}}$ -axis and the spin of B along the $\mathbf{\check{b}}$ -axis. Let’s see what happens if we decide to measure the spin along the x-axis: $\mathbf{\check{a}}=\mathbf{\check{b}}=\check{x}$ . For 1/2-spins, the spin operator $S_{A/B}^{x}$ can be represented by the 2×2 hermitian matrix

$S_{A/B}^x=\frac{1}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ . (2)

By performing a change of basis we can rewrite the state $|\psi\rangle$ in terms of the eigenstates of (2)

$|u\rangle=\frac{1}{\sqrt{2}}(|\!\downarrow\rangle + |\!\uparrow\rangle), \qquad |v\rangle=\frac{1}{\sqrt{2}}(|\!\downarrow\rangle - |\!\uparrow\rangle)$ . (3)

Using Eq. (3), we can rewrite the state (1) as

$|\psi\rangle=\frac{1}{\sqrt{2}}(|vu\rangle-|uv\rangle)$ .

Just like before, by choosing to measure the spin of A along the x-axis we can determine it’s value and infer the value of the spin of particle B without the need to measure it (and vice versa). Furthermore, it turns out that this is the case independent of the election of the axis we choose to measure! (Provided that $\mathbf{\check{a}}=\mathbf{\check{b}}=\mathbf{\check{v}}$ ).

This is exactly the same situation as the one we discussed in this article, a simple choice of the axis along which to measure the spin A allow us to establish the value of the spin of B along this same axis without the need to measure it. And this is also the case (as we already saw) for physical properties described by non-commuting operators ( $S^x$ and $S^z$ do not commute). Thus, according to EPR’s criterion of reality, these physical properties are elements of reality.

This apparent paradox was finally solved by an experiment proposed by physicist J. S. Bell. If you are interestes so far, I recommend you read my article to find out how this all turns out.

EPR’s Paradox Exemplified: Bohm’s Spin Experiment by Marco Vinicio Sebastian Cerezo de la Roca is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.