In order to understand the EPR paradox we will consider a simple example based on the Quantum Mechanical (QM) idea of spin angular momentum (or just spin). It would be prudent, then, to first introduce and try to wrap our heads around some of the weird properties of Quantum spin.
Quantum Spin
Spin, in the classical sense, is nothing new: The Earth spins, tops spin, etc. In the Quantum world, however, we can't think of spin in the same way. This is because, in QM, the particles which we are discussing are often fundamental building blocks of matter (e.g. electrons). Fundamental in the sense that they are point particles, indivisible and with no insides. When something spins, in the classical sense, it rotates about some point within itself. A point, though, has nothing within itself about which to rotate!
So if the point particles don't actually spin, why give them a property with that name? Well, although the actual motion of the particles is certainly not one we would recognize as classical spinning, there are subtle similarities between the classical and quantum notions of spin. When a charged, macroscopic ball spins it creates a magnetic field. Specifically, it becomes a magnetic dipole with a North and a South pole. This is because of Maxwell's Equations (specifically, Ampere's Law) which state that movement of electrical charge (i.e. current) creates a magnetic field. Charged quantum particles, like the electron shown below, create magnetic fields just as if they were macroscopic balls of charge spinning in the classical sense. Thus, we at least have some kind of justification for calling spin spin.
The next weird thing about spin in QM is perhaps not very surprising: it is quantized. We will be thinking mostly about electrons throughout so we'll use them as our example here. Electrons have a certain value of spin, call it a. We can set up an experiment to measure the component of the spin in a certain direction. Our classical intuition leads us to expect that our measurement might return values in a continuous range from -a (spin pointing anti-parallel to axis of measurement) to +a (parallel). In other words, we expect to get all kinds of results: -a/4, -a/8, +7a/32, whatever. In fact, though, what we find is that only two values are ever measured: -a and +a. Nothing in between. That is what I meant by spin being quantized: When measured, it can only take on one of two values instead of any value in the continuous range from -a to a, as we would expect classically. Bizarre? Yes. But it's what puts the Quantum in Quantum Mechanics.
Now that we have some idea what spin is, we can begin to develop and understand the EPR paradox and the thought experiment that spawned it.
The EPR-Bohm Experiment
Consider a particle sitting around doing nothing. A pion, say. Our pion suddenly decays to an electron and a positron (i.e. anti-electron), as it is wont to do. The pion has no spin angular momentum (no pions do) and, let's say, no other angular momentum. Armed with the knowledge that total angular momentum must be conserved in the decay, we deduce that the net angular momentum of the electron and the positron must also add to zero. Now, let's set up some detectors, one to detect the spin of the electron and one for the spin of the positron. See the image below (taken from Griffith's brilliant Quantum book). We'll suppose that the two detectors measure spin along the same axis, which we call the z-axis.
Say the measurement of the electron returns the value +a. We then immediately know, even without a direct measurement, that the spin of the positron must be -a so that the total adds to zero. That's, essentially, the EPR paradox. Seems a bit anticlimactic, doesn't it? Well, it's not. I'll try to convey why in the next post.
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