This is about identical particles, Bosons and Fermions, and why the world seems solid when we kick it and how physics tells us that everything is really one thing.
Alice and Bob are both people, but you would never confuse them—they’re not identical. “Identical twins” are a lot closer. Some people can’t tell them apart, but their mother can, their spouses can. Imagine opening a box from the Acme Ball Bearing Company, picking out two stainless steel balls that look exacly alike. You really can’t tell them apart. Maybe looking for pits and scratches with a microscope, or with a scale that weighs accurate to a nanogram you might be able to distinguish them; but even if you couldn’t tell them apart at all, you would always think it’s a meaningful question to ask, ‘is this one ball number one or ball number two?’
We think of two electrons or two photons (light particles) with an extension of our common sense from larger objects. They have no nicks or scratches, and even the most sensitive measurement apparatus can’t detect a difference in their mass or any other properties. But we think of them as distinct in principle. You, me, and the physics professor, we all think that this electron is the one over here, and that is the one over there. In fact, micro manipulation technology has allowed us to line up individual atoms in a pattern, and they stay put!
But (you knew there was a “but” coming after this long-winded explanation of what everyone knows) the formalism of quantum mechanics tells a very different story. Actually, it’s two very different stories, very different from what we imagine and very different from one another. These are the stories of Bosons and Fermions.
Light is made of photons, which are Bosons. The equation that governs the movement of a collection of photons, is explicitly symmetrized to account for the fact that the photons are identical in principle. What I mean by ‘symmetrized’ is that you write down the equation with photon #1 over here and photon #2 over there, then you write the same equation with photon #1 over there and photon #2 over here, you add up the equations and divide by two. So, what difference does it make to go through this Chinese fire drill, averaging up the two terms that are really no different from one another? Why bother? Yes it makes a difference, it makes a big difference. Technically, it’s because the wave function for the photons is squared before you evaluate the intensity of the light, so that factor of 2 in the average doesn’t come out in the wash. The probability is twice is high for two photons to be coupled together, moving in lockstep, acting like one big photon. And for a million photons, the probability is a million times as big. Once you get a large number of photons all moving in lockstep, the probabilty for the next photon to join them is very high. The upshot is that lasers are possible. Just a tiny crystal and an LED is enough to line up the photons, all moving in lockstep. (When I was in college, a laser cost $100,000, but now they’re cheap enough that people buy them just to amuse their cats.) Lasers work because of the way that identical Bosons behave in quantum mechanics.
Electrons are Fermions, the opposite of Bosons in quantm lingo. Instead of adding up their wave functions and dividing by two, you subtract them and divide by two.
[particle #1 here, particle #2 there] – [particle #2 here, particle #1 there]
This minus sign is entirely responsible for the illusion of separateness which is so deeply embedded in our perception of the physical world. It’s because for the special case where “here” and “there” are the same place, the two terms cancel out and the probability is zero. For two Bosons to be in the same place at the same time, the probabiliy is doubled; but for two Fermions, the probability is zero. Never happens. (This is called the Pauli exclusion principle.) And as the two Fermions get too close together, they start to sense this and they get antsy. They can only be brought very close together by giving them a great deal of energy.
When “here” and “there” are far apart, we imagine that particle #1 is over here and particle #2 is over there. Applying our experience from everyday life, we think of them as separate and distinct, but the QM equations are telling a different story. Both electrons are here, and simultaneously both electrons are there, and the two electrons are conspiring to keep a distance between “here” and “there” — not because they are different electrons, but precisely because their fates are locked perfectly together in this weird way, with the minus sign.
So, why is the rock hard when you kick it? Maybe you thought that the atoms in your shoe take up space and the atoms in the rock take up space and they can’t be in the same space at the same time. That would be wrong. Or maybe you thought the electrons in your shoe have a negative charge and the electrons in the rock have a negative charge, and the two negatives strongly repel each other when they get close together. That’s a very educated guess, but it’s also wrong.
The reason that the electrons in your shoe and the electrons in the rock kick up a fuss when they are in very close proximity is that the electrons in the rock and in your shoe are, at the most fundamental level, the same electrons, part of the same wave function. The lowest energy state of that electron pair has two lobes, with empty space between them, and unless you have a whole lot of energy to bring those electrons up to the next higher energy state, they are going to conspire to maintain that empty space between them.
But there are more than two electrons in the world
How would you write the wave function for 3 or 4 or 1080 electrons? This part gets technical, but I’ll write it down for those who find it fun.
Let’s say there are 3 electrons and three different places. Call the places One, Two and Three, and call the electrons 1, 2, and 3. Then
means that the 1st electron is in the first place, the 2nd is in the second place and the 3rd is in the third place. There are 5 other possibilities. For example, Electron 1 can be in place two and electron 2 can be in place one. Electrons 1 and 2 have swapped places. Every time that happens, there’s a minus sign. If there are an even number of swaps, then there’s a plus sign; odd number of swaps contributes a minus sign. The wave function has a structure like this.
One(1)Two(2)Three(3) + One(2)Two(3)Three(1) + One(3)Two(1)Three(2)
– One(3)Two(2)Three(1) – One(2)Two(1)Three(3) – One(1)Two(3)Three(2)
Yes, physicists really do work with combinations like this, and they have a name: they’re called Slater determinants. These algebraic expressions have the property that if the functions One(), Two(), and Three() are well-separated in space then the pluses and the minuses reinforce to produce a large amplitude (probability); but if there is significant overlap, the pluses and minuses tend to cancel so the probability is close to zero.
With just four electrons, there are 12 positive terms and 12 negative terms. The number of plus and minus terms in the Slater determinant increases very rapidly — I want to say increases exponentially with the number of electrons, but that would be an understatement. The number of combinations is much bigger than that. There are 1080 electrons in the universe and their wave function is a Slater determinant with 1080! terms. That’s “1080 factorial” which is the biggest number you’re ever likely to come across in a discussion of one universe.
Quiz for the truly nerdy: How big a number is 1080 factorial? If you tried to write it down, how many digits would you have to write? Is it bigger or smaller than a googol? Is it bigger or smaller than a googolplex?
|We talk about a particular electron being in a particular place, or following a particular orbit. But this is a shorthand, a fiction. The truth is that every electron in the universe participates equally in this behavior, whatever it is, and all the electrons are continually checking in with each other and coordinating their behaviors, such that if you shine a light on this place and look, exactly one electron will appear under your flashlight. Every one of the 1080 electrons in the universe contributes to the probability of your detecting an electron there and then, and no meaning can be attached to the question, “which one was it?”