# John Bell’s Inequality Theorem of Quantum Mechanics

FROM THE LECTURE SERIES: THE EVIDENCE FOR MODERN PHYSICS: HOW WE KNOW WHAT WE KNOW

## A question that scientists face is whether it is possible to distinguish between the traditional and interconnected theory of quantum mechanics, where particles are simultaneously in multiple states and they pick a state when they are observed, and the other possibility, where some version of hidden variables applies, and there is something going on that we haven’t detected. Is there an experiment that could distinguish between the two? John Bell’s Inequality theorem was a significant contribution to the concept of quantum entanglement. (Image: Ezphoto/Shutterstock)

### Bell’s Inequality Experiment

It turns out that there is, and it was first written down in 1964 by a physicist from Northern Ireland by the name of John Bell. What he did was to show that a hidden variable theory of quantum mechanics was not possible if the speed of light was to be respected. His theorem is called Bell’s inequality.

We can directly focus on the experiment that is predicted to give different results for hidden variables and ordinary quantum mechanics. It goes something like this. Suppose we replace the plus and minus idea of spin with up and down arrows instead. They really are the same concepts, with, for example, plus being up and down being minus. But the up and down arrow concept is a bit better because we can orient the arrows.

We can point them in any direction as long as the two arrows are in opposite directions, so if our left particle has an up spin, the right particle has a down spin. If the left particle has a spin arrow pointing left, the spin arrow for the right particle is pointing right. If the left particle has a right arrow, the right particle has a left arrow, and so on.

Now, suppose we have two particles with opposite spins as was described so far. Before the observation, both particles are in both spin states. Then, when you do the observation, one spin configuration is observed. We will do the experiment many times.

### Predictions in the Experiment

Now, imagine that each time we measure the spin direction of the one on the left and only keep the ones in which the spin is up. We then look at the particle over on the right and ask what fraction of the time the spin is in a variety of directions. Classically, it’s easy. If we ask what fraction of time we’ll measure the spin of the particle on the right to be up, and we know that happens 0% of the time.

If we ask what fraction of time the right-hand particle has a spin downward, it’s 100% of the time. And we get the same predictions for both quantum mechanics and for hidden variables.

The situation is harder when we ask what fraction of the time we’ll measure the spin arrow of the right hand particle being to the right or left. Classically, that never happens, but the situation is more complicated when you do a full quantum calculation.

Quantum mechanics predicts that we’ll find the spin of the right-hand particle to the right, meaning 90° from the vertical 50% of the time and to the left, meaning at 270°, is also 50% of the time. As it happens, the theory of hidden variables makes the same predictions.

The situation is more complicated when we ask how often we find the spin of the right-hand particle at an angle of 45° from the left-hand particle. In quantum mechanics, it’s very rare, but not zero. If we ask what fraction of the time we see the right hand particle having a spin pointing 45° away from straight down, we see it’s pretty often, but not 100% of the time.

And if you ask the same question for any orientation of the spin of the right hand particle compared to the spin of the left hand particle, we see a quantum prediction that is a smooth undulating curve. If we ask the same thing for hidden variables, we see that the prediction is a little different. It’s a series of straight lines like a saw tooth going up and down.

This article comes directly from content in the video series The Evidence for Modern Physics: How We Know What We Know. Watch it now, on Wondrium.

### Hidden Variable Theory It may be possible to distinguish between the hidden variable theory and quantum mechanics. (Image: Atdigit/Shutterstock)

So, now, we have a prediction that distinguishes between the hidden variable theory, which says that somehow the two particles know what the other one is doing, but they don’t communicate at faster than light, and ordinary quantum mechanics, which allows for the wave function to collapse and the information to transfer faster than light.

So, has the experiment been done? Well, actually, yes. French physicist Alain Aspect did the experiment in the early 1980s, not that long ago. He measured the probability of finding the second particle at all possible spin directions and he found, quite definitively, that point after point, the data agreed perfectly with the predictions of quantum mechanics and not hidden variables.

### Difficulty with Quantum Mechanics

We know that the rules of quantum mechanics are an accurate description of reality. If you connect two quantum objects and then separate them by great distances, they remain inextricably linked in ways that just don’t seem to make sense. And, when you get right down to it, we don’t understand what’s going on at a deep and intuitive level.

The luminaries of the 1920s who invented quantum mechanics thought about it and failed. For nearly a century now, thousands of physicists, physics students, and physics enthusiasts have thought hard about what quantum mechanics is telling us and have come up short. Even the great Richard Feynman said in his lecture and writings on the subject, “I think I can safely say that nobody understands quantum mechanics.”

Even if we don’t understand the theory fully, scientists have used it to revolutionize the world. Our modern technological world depends crucially on quantum mechanics, from lasers to microchips, from LEDs to the burgeoning technologies of quantum computing and quantum cryptography.

### Common Questions about John Bell’s Inequality Theorem of Quantum Mechanics

Q: What is John Bell’s inequality theorem?

John Bell’s inequality theorem shows that a hidden variable theory of quantum mechanics was not possible if the speed of light was to be respected.

Q: What were the findings from Alain Aspect’s experiment?

Physicist Alain Aspect did an experiment in the early 1980s where he measured the probability of finding the second particle at all possible spin directions and he found, quite definitively, that point after point, the data agreed perfectly with the predictions of quantum mechanics and not hidden variables.

Q: How are scientists using quantum mechanics today?

Scientists are using quantum mechanics to revolutionize the world. Our modern technological world depends crucially on quantum mechanics, from lasers to microchips, from LEDs to the burgeoning technologies of quantum computing and quantum cryptography.