I want to start off by telling some stories about things that can happen to electrons. I guess it's fair to say that I think these are among the most unsettling and surprising stories to have emerged from the natural sciences. One of the properties whose measurements we're going to be talking about here, let's call the color of these electrons. Once again, electrons don't actually have colors. I'm talking about certain components of the intrinsic angular momenta of these electrons.
To keep it simple, let's call one of these properties the color of the electron. And it happens to be a feature of electrons that when you measure this property, there appear to be only two possible numerical values that this color property could take on, say plus one or minus one. Let's call one of them black and the other one white. So it turns out to be the case that every electron whose color you measure either turns out to be a black electron or it turns out to be a white electron. You never see green electrons or purple electrons or yellow electrons or anything like that.
Another of the properties we're going to be measuring, we're going to be talking about measurements of is we'll call the hardness of the electron. That only has two possible numerical values as well. Let's call one of those values hard and the other one soft. Good. So like I said, it turns out, well, the first thing to say is these are properties that we've known how to measure in a very routine way for something on the order of 100 years now.
We're very good at measuring these color and hardness properties in the laboratory. The way you measure them is by arranging a certain configuration of magnetic fields. There's a certain configuration of magnetic fields which will deflect an incoming white electron in one direction and will deflect an incoming black electron in a different direction and similarly for hard and soft electrons. So I'm going to refer to these arrangements of magnetic fields as boxes, as measuring boxes. So it's routine.
It's easy to construct in the laboratory something that you might want to call a hardness box. This is a box with three apertures. You feed an electron into this box. If the electron you feed in is a hard electron, then the magnetic fields on the inside arrange things so that the electron exits the box by the hard aperture. If the electron you feed in is a soft electron, then the effect of these magnetic fields on the inside is to shoot the electron out through the soft aperture.
We can build in a very similar way color boxes. You feed an electron into the input aperture of a color box. It comes out this aperture if it's a black electron and it comes out that aperture if it's a white electron. Okay. Good.
Like I say, we've been very good for something on the order of a century now at measuring these properties, at manipulating electrons with these properties, at feeding them into the boxes we want to feed them into, so on and so forth. Once you have boxes like this, one thing that might occur to you right away once you've discovered these properties, learned how to measure them, and by the way, it's going to be important to a lot of what we're going to say here that these measurements are repeatable in the way you would expect a measurement of a bona fide physical variable to be. That is, if I measure the color of an electron and I find it to be white and I immediately feed it into a second color box, 100% of the time it'll come out the white aperture of that second color box. And if it came out the black aperture of the color box and I feed it immediately into another color box, it'll come out the black aperture of that color box. There are things you can do to electrons, as we're going to see, which can affect their color values, okay?
So it's not the case that any electron that's ever come out a white aperture is going to come out the white aperture of any color box you feed it into at some later time in its life. But if you keep the environment free of things that could disrupt these color and hardness values, if you repeat two measurements in a row, you'll always get the same result. If that weren't true, we would begin to be puzzled about what we even mean by speaking of ourselves as measuring a physical variable when we carry out a measurement like that. Okay. We have these, we're able to move these electrons around as we wish.
We're able to build these color boxes. Something that it might immediately occur to us to do is to ask, okay, we have two physical properties of electrons that we know how to measure. Is there any relationship between these properties? Are the physical properties Connected with one another in some way. An easy way to find out if that's true, or at least to begin to get a handle on whether or not that's true, is to look for statistical correlations between the color value of a given electron and its hardness value.
So we can measure the color values of a whole bunch of electrons, measure their hardness values, and look or dump the results into a computer and ask the computer to look for correlations between those values. These experiments are easy to do, and it turns out when you do these experiments, there are no correlations whatsoever. That is, it turns out of any large collection of electrons that are, say, known to be white, statistically speaking, exactly fifty percent of them turn out to be hard electrons and fifty percent of them turn out to be soft electrons. And similarly for all possible combinations. Um, so there appear to be no correlations at all between the color values of a given-- the color value of a given electron and the hardness value.
Good. Um, here's another experiment that it might occur to you to do, um, if you have these color boxes lying around in the lab. Um, um, I don't know if it would occur naturally or not, but here's an experiment that people do. Suppose I set up a sequence of three boxes. So I have, say, a hardness box here, followed by a color box.
Okay, so here's the hard aperture, here's the soft exit aperture, here is the black exit aperture, here's the white exit aperture. And I follow it up with another hardness box. So here's the hard aperture, here's the soft aperture. Good. Um, suppose I do the following.
I feed a stream of electrons in here. Uh, the ones that come out the hard aperture, I throw them away. I only save the ones that come out the soft aperture, and those I feed into the color box. Okay? Um, half of those statistically will come out the black aperture, half will come out the white aperture.
Throw away the ones that come out the black aperture, hang on to the ones that come out the white aperture, and feed those into the intake aperture of this second hardness box. Well, by the time the electrons get here, they've all been measured to be soft here, and they've all been measured to be white here. Um, and so it seems natural to suppose that what we're dealing with by the time we get here are electrons which are all soft and white. Okay? And so our expectation would be that feeding them into a second hardness box is just gonna confirm what we already know about these electrons, which is that they're soft.
So we'll expect them all to come out this aperture. And the first minor surprise in this story, um, is that that's not what happens when you do this experiment. Um, if you take a bunch of electrons here, all of which have come out the soft aperture of this hardness box and all of which have come out the white aperture of this color box, when you feed them into a second hardness box, the statistical facts are that half of them come out the soft aperture and half of them come out the hard aperture. So you experiment around a lot with lots of different designs for this color box with more and more careful engineering of the color box, so on and so forth. Um, and the impressive result here is that it's not merely the case that you don't succeed in building a color box which leaves the hardness values intact.
Okay? Um, it's much more than that. You find that no matter what you do, you're-- as long as what you end up with is a box that, that does its job of measuring colors, okay, as long as that's what you end up with, you seem to be stuck with a box that flips the hardnesses of, statistically speaking, exactly fifty percent of the electrons that pass through it. Okay? No amount of engineering, insofar as we can tell, no amount of redesign of this color box, as long it re-- as it remains a device that succeeds in measuring the color, nothing we can do seems to move those fifty/fifty statistics a thousandth of a percent off of exactly fifty/fifty.