# Bell’s Inequality

John Bell was a smart dude. He looked a little like the guy from Breaking Bad. Who, on a side note, decided to the use Heisenberg for his alias in the show. I think he should have used Bell personally since he looked like him).

The Problem
The 2nd probability I explained in the previous post was hard to swallow by a lot of people, including Einstein. So much in fact that Einstein got together with a few students of his (Podolsky and Rosen) and wrote a paper outlining that this kind of probability was nonsense and there must be something that’s just not being measure (A hidden variable). The paper is referred to as the Einstein, Podolsky, Rosen paper (EPR). The paper itself is long and I’ve never actually read it all the way though. I’m not smart enough. But here’s the gist of it as I understand it.

EPR
Heisenberg’s Uncertainty principle says that we cannot know the position and momentum of a particle with 100% certainty. Mathimatically it’s shown as the following if $x = position$, $p = momentum$ and $\hbar = Plancks's Constant$

$\Delta{x}\Delta{p} \approx \hbar$

The EPR attacks this directly. The EPR proposed the following.

Suppose two particles (A and B) are shot at each other at very high speeds. These two particles would pass so close to each other, that they would pick up their traits. Now the two particles are zipping away from each other with the exact same momentum and position in relation to the point where they passed each other. Now take a measurement of particle A’s momentum and position. This would be an indirect and exact measurement of particle B’s momentum and position.

This seems pretty logical if you’re thinking about the problem from a Newtonian point of view. What a quantum theory point of view would suggest is that as soon as you make a measurement on particle A, particle B would then alter it’s own properties as if you had measured it directly.

So we’re left with two possibilities. Either Einstein is right and there is a hidden variable that we don’t yet understand, or the quantum physics is correct and there is a spooky “action at a distance” as it where to be later explained.

The Solution
So arises the question… “How do we test this?”. John Bell came up with a way do this. He started by outlining a theoretical experiment. This experiment involves sending two entangled quantum particles down two separate mediums, and filters (polarizer) at then end of these mediums. After the quantum particle passes through the filter… they could then be measured by a coincident counter what the resulting particles properties are. This could then tell them with if the particles are just acting by themselves (hidden variable) or if they influence each other (quantum entanglement).

Bell came up with a very simple inequality and graph to use.

$1 > x > -1$

He suggested that the experimenters do the experiments and plot it out. If the plot looks like the straight lines on the graph, then that would suggest there is a local hidden variable. If the plot took the shape of tiny curves, then that would suggest that there is an action at a distance and two particles are “entangled”.

This experiment has been done multiple times over history and the majority of the time, it appears that quantum entanglement is the winner.

Conclusions
It’s hard to make a conclusion in quantum physics. It’s a field that rarely gives “answers”. It seems like it just gives information in the form of more questions, which some could argue is true about most studies in science and specifically in theoretical physics. You also have to be careful not to get in the habit of trying to “figure out what’s going on”. You can’t picture it in your head like you can with classical Newtonian physics. All you can really do is show what the data is telling you. And in this case, the data is suggesting that making a change to one particle can and will instantly affect the other particle if the two are entangled. This is kind of crazy stuff and really challenges our ideas of what a particle really is. Leading to more questions, more math, and more experiments.

Category(s): Jared's blog, Quantum Electrodynamics, Quantum Physics