How do we know that all electrons are identical? Part 1
You may have heard physicists say “all electrons are identical”. But what does that mean? How could we ever know that, if they are so small that nobody has ever even seen one directly? And even if we do have some way of knowing that, why does it matter?
When it comes to any kind of thing that we can’t see directly, we tend to conceptualize it by analogy with things that we can see directly. For example, we can see cars directly. And we know from experience that even though most (say, for example) 2015 Toyota Camry’s are nearly identical, we also know that there are little differences here and there between the particular instances of them. Even if they were all manufactured by the same process, you might find one that has a minor scratch on the front right bumper, another that has a tiny dent near the left rear wheel, etc. By analogy, it seems natural to assume that since we can’t see electrons well enough to inspect every single one of them carefully, there could be one here or there that has a scratch or a dent. Even if most of them are nearly identical, it sounds like a pretty arrogant and far fetched thing for a physicist to claim that all of them are identical.
But this reasoning is based on a false analogy between macroscopic objects like cars, which behave classically — and microscopic objects like electrons, which behave quantum mechanically. Even the word “object” is a pretty poor choice of words to describe an electron. An object is usually something you can hold in your hand and touch and feel. You can’t touch, feel, taste, see, or smell an electron. An electron is something more abstract and less tangible like the hour of a day or a personal checking account at a bank. It’s real enough in that it has an impact on our lives and we wouldn’t be able to paint a convincing picture of the reality we live in without introducing some concept of it into our language; yet it’s not something that can directly affect any of our senses. Usually, we only start to notice their effects on the human scale if you gather enormous numbers of them together into certain configurations, such as in atomic orbitals or electrical currents. (See, for instance, the visually striking image of lightning at the top of this post. If this were only a single electron instead of an astoundingly large number of them — we wouldn’t notice any pattern in the sky.)
Now that I’ve got you hooked, I’ll let you in on a little secret… this post is not actually about electrons specifically, it’s about any kind of microscopic particle. Electrons are just one of the simplest well-known examples, so it’s often picked to illustrate the principle of identical particles. The truth is, all subatomic particles of a particular kind are identical with each other. That is, all electrons are the same as each other, all protons are the same as each other, all neutrons are the same. It doesn’t even stop there. What about atoms themselves? Now we have to be careful, because there are different kinds of atoms: Hydrogen, Helium, Oxygen, Lead, Neon, Uranium. Each type of atom has a different number of protons, electrons, and neutrons. But once you’ve specified the total number of these 3 components an atom has, individual instances of that type of atom are identical and indistinguishable.
At this point I should clarify more what I mean by identical. Yes, the same type of atom can be in different states. For instance, imagine you have a sample of the simplest kind of atom — a Hydrogen atom with one proton and one electron (also known as ¹H or “protium” to distinguish it from other isotopes of Hydrogen like deuterium or tritium which include neutrons). Most of the Hydrogen atoms could be in their ground state, but some might be in an excited state. If they are in different states, then that does give us a way of distinguishing them. But this kind of distinguishability is not what is meant by “identical particles”. All ¹H isotopes are identical in the sense that they each have the same set of states they could be in, and there are no intrinsic properties which distinguish one from another aside from which state they are in. (Part of the state of a particle is its location, so that’s another way to distinguish them.)
This may not sound that weird, but if you think about it, it’s very different from how macroscopic things work. One person could be in a state of excitement, while another could be in a state of sadness or confusion; or one person could be in Virginia and another could be in Texas. But presumably, in addition to all of the different state information like this that could be used to tell the two people apart, there would also be some intrinsic differences between the two people. Maybe one is 6'2" while the other is only 5'4". One has dark skin and the other light skin, or one has freckles and red hair. These are fixed properties associated with them that stay with them always, they don’t change as their state changes. These kinds of fixed properties of a person make up their “identity”. People have individual identities, but electrons and other elementary particles do not. Instead, they have different temporary states they can be in and the potential states they could be in at any moment are the same as any other particle of the same type. Nothing else serves to distinguish them.
How do we know this? The first big hint of it — believe it or not — came from entropy. Apologies in advance that some of this post is going to get a bit more mathematical than most of mine. But if you can make it through, you will have understood a lot more about what entropy really is than the vast majority of people do, including many engineers and scientists.
The information contained in any system is determined by how many states it can be in. For example, a fair coin can be in 2 states after it lands: heads, or tails. That means it takes 1 bit of information to describe its state (2¹ = 2). If you have 1000 coins, then they can each be in 2 states, and the total number of states is 2¹⁰⁰⁰. Therefore, it takes 1000 bits of information to describe this 1000-coin system. The amount of information in the system is a logarithmic function of the number of states.
What physicists call the “entropy” of a system is a measure of how much of the information in a system is unknown, too difficult to measure, or just not worth keeping track of. For example, in the case of the 1000 coins, let’s say that you know 654 are heads and 346 are tails. But for some reason you don’t know or care which ones are heads and which ones are tails. Some of the information is known, while some of it is missing/hidden. This hidden information is called “entropy”.
The maximum entropy that the 1000-coin system can have is when exactly 500 are heads and 500 are tails. That’s because there are many many different states where this is the case (2.7 x 10²⁹⁹ states, in fact — more than the total number of atoms in the known universe!). So even though you know how many came up heads and how many came up tails, you still know very little about what each coin is — it would be hard to guess if you pointed to a coin at random whether it’s heads or tails — you just don’t have enough information about the state of the coins.
In contrast, the minimum entropy the 1000-coin system can have is when either all coins are heads or all coins are tails. In either case, there is only 1 unique state where this happens. In terms of probability, imagine that you flip the 1000 coins all at once. You know that you’re extremely unlikely to see every single one of them come up tails, or every single one of them come up heads. But you’d be a lot less surprised if 500 of them landed on heads and 500 landed on tails. In fact, if you don’t see something at least close to that happen, you should be seriously wondering if the coins you used are really fair (evenly weighted)! This is one of the simplest illustrations of the 2nd law of thermodynamics I can think of: without rigging the coins in any way, the system will naturally wind up in a state of near maximum entropy.
Often physicists use the term “macrostate” to mean the state information that is easily accessible and trackable by humans, and “microstate” to mean all state information about a system — including the hidden information that lurks in the microscopic details that humans ordinarily don’t know or care about. In the example of the coins, the macrostate can be represented just by a single integer between 0 and 1000, the total number of coins that have landed heads. While the microstate would have to be represented by a binary string 1000 digits long like 10001011101111111101110010101011… (I stopped typing after 32 digits, but imagine I kept going for another 968 digits). Obviously, there is a lot more information contained in the microstate than in the macrostate.
In the case of a collection of particles, such as an “ideal gas”, the microstate is determined by the individual positions and velocities of all of the particles, while the macrostate just contains information about the total energy of the system, and its temperature, pressure, and volume. The amount of information hidden in a typical microstate is astronomical compared to the small amount of information summarized by the macrostate — even though for most physical systems, all anyone cares about is the macrostate. (Because the macrostate is what affects outcomes in the human world directly.) Change around the individual positions of a few particles in a gas, and not much is different — nobody notices — but change the temperature or pressure of a gas, and suddenly that’s something that could have a big impact on the every day world we live in!
Imagine we have some Hydrogen gas. Before the discovery of quantum mechanics, it was often assumed that the microstate of a gas of particles was different from a similar state where 2 or more of the particles swap places with each other. In other words, it was assumed that Hydrogen atoms are at least in principle distinguishable from one another. This assumption affects the count of the total amount of information it requires to specify the microstate.
In the late-1870’s, Josiah Willard Gibbs pointed out a paradox in statistical mechanics based on the classical understanding of particles at the time — it became known as the Gibbs paradox. Imagine you have 2 boxes filled with different isotopes of Hydrogen. In the diagram to the left, the red dots could represent ¹H (each a proton and an electron) and the green dots could represent ²H (each consisting of one proton, one neutron, and one electron — also known as deuterium).
The Gibbs paradox had to do with the question of whether the total entropy of a system is equal to the sum of the entropies of each part of a system. This was seen as an important axiom early in the development of thermodynamics — without it, one could devise experiments which would violate the 2nd law. But by the late 19th century, this had run into conflict with the assumption that the individual particles of a gas are distinguishable from each other.
Initially the door between the chambers is closed, and the left box is entirely filled with red dots while the right box is entirely filled with green dots. If the total number of dots in each chamber is N, and the number of possible states each dot can be in is n (counting all different possible positions and velocities it could have within one of the two chambers), then naively the number of states in each chamber would be n^N. Since both chambers are independent from each other, the total number of microstates the system as a whole can be in is just the product of the states in each chamber: (n^N)x(n^N) = n^(2N). This is part of the basic rules of probability and combinatorics — if the probability of finding the left box in a particular state is p = 1/(n^N) , and the probability of finding the right box in a particular state is also p = 1/(n^N), then as long as the two are independent from each other you can just multiply the two probabilities to get p² = 1/(n^(2N)) for the probability that the entire system is in a particular state. Entropy is a logarithmic measure of the number of microstates corresponding to a given macrostate. Because the number of microstates for the entire system is given by the square of the number of microstates for each half of the system, that means the total entropy of the system is exactly twice what it is for each half. (Logarithmic measures turn multiplication into addition, for the same reason it takes 5 bits of information to describe a system which has 2⁵ states, and 5+5 = 10 bits of information to describe a system which has 2⁵ x 2⁵ = 2¹⁰ states.)
But what if the two boxes are not independent? We can achieve this by opening the door in the middle wall that keeps the two chambers separated. If the red and green dots are moving around entirely randomly, as gas molecules tend to do, then gradually the two types of Hydrogen gas will begin to mix with each other. As this happens, the entropy of the entire system will increase, until it eventually reaches the state of maximum entropy where the two gasses are fully mixed. Since at that point, each of the dots — red or green — can be anywhere within either of the two chambers, there are now (2n)^(2N) = 2^(2N) x n^(2N) states total instead of just n^(2N) before the door was opened. The extra factor of 2^(2N) that the number of states have become multiplied by means that the total entropy has increased by an extra 2N bits.
Where did these extra 2N bits of entropy come from? This is the extra hidden information that could be used to tell which of the 2 chambers each dot exists in. Because there are 2 possibilities for each atom, that’s 1 bit required for each atom. And since there are 2N atoms total (N red and N green), that’s 2N bits of information.
So far, no problem. The entropy is lower when the gasses are each isolated in their own separate chambers, and then increases when they mix together. If you close the door in between the chambers, the gasses remain mixed and the entropy increase cannot be reversed. Pretty standard thermodynamics. But what happens if instead of ¹H in the left chamber and ²H in the right chamber, we had started with all of the same gas (say, ¹H) in both chambers?
We can make the same argument as before. The number of states when they are both isolated is still n^N in each chamber, so n^(2N) for the whole system. Open the door in the middle, and now there are 2n different gas molecules which can be in 2N different states, so our n^(2N) states has increased to (2n)^(2N) states. Again, this is an increase of 2N bits of hidden information. The extra information is information that specifies which chamber each of the atoms is in, as they all started out in a fixed chamber from which they couldn’t escape. Simply stated, opening the door allows for more possibilities —based on the math, it doesn’t seem like it matters whether it’s possible to distinguish which particles came from which initial chamber as it is in the case where some were labelled red and some green.
But now we encounter the problem, and it’s a big one. If we started out with equal amounts of ¹H gas in each of the two chambers, we’ve concluded that opening the door should increase the entropy of the system. But what happens if we then close the door, restoring the system to its original state where we again have equal amounts of ¹H gas in each of the two chambers? If the entropy doesn’t go back down, then we’ve somehow managed to get back to the same macroscopic state but with higher entropy. And if the entropy does goes back down, then we’ve found a physical operation that can reduce the total entropy of the system without requiring any work — something that goes completely against the 2nd law of thermodynamics!
This is Gibbs’s paradox. The basic problem here is that if we think about the system after the door is opened, then there are at least two different ways of counting up the total number of possible states of the system (the logarithm of which gives us the entropy), and yet the two different methods don’t agree with each other. Since both halves of the system are similar to each other, one way is to count the states in each half of the system and then square that to get the total. If you do this, you get n^N in each half which means n^(2N) total. But if you instead think about the system as a whole, you get (2n)^(2N) states which is equal to 2^(2N) times the first answer. If the concept of entropy is to make any sense at all, then these two answers should agree with each other!
In the case where two different kinds of gas mix, it makes sense to think that the entropy increases after we open the door, because there a fewer possible ways in which you can arrange green balls in one chamber and red balls in another chamber, as opposed to green and red balls all mixed together homogeneously. It seems that the world has become less organized after the mixing happened, and the entropy increase is needed in order to account for that. But there shouldn’t have been any entropy increase if all of the balls were indistinguishable. Therefore, for indistinguishable particles we have to modify our naive way of counting states a bit.
The only way to resolve the paradox, and to ensure that entropy makes sense as a concept, is to treat all states which just involve a permutation of identical particles as the same state. By a “permutation” I mean swapping the states of two identical particles with each other. For example, imagine that particle A is an atom in the left chamber at location x1 with velocity v1, and particle B is an atom in the right chamber at location x2 with velocity v2. If we swap particle A with particle B but leave all of the other particles in the gas the same, does that count as the same state for the overall system? The right answer is: if they are distinguishable (for instance if A is an ¹H atom and B is an ²H atom) then the overall state is different, while if they are indistinguishable (for instance if A is an ¹H atom and B is also an ¹H atom) then the overall state is the same. In the latter case, computing the right value for the entropy depends crucially on avoiding what’s known as the “double counting” of identical states. Double counting means counting the state AB as different from the state BA. This is a serious no-no; they must both be counted only once as they are the same state. Otherwise, the laws of thermodynamics would just all crumble and fall apart!
So what is the correct formula for the number of states in a system? Where did we go wrong? We went wrong right at the beginning, with the assumption that the number of states in each chamber was n^N. This would be true if particles of the same type were distinguishable, but for identical particles you have to correct for the problem of double counting by dividing by a permutation factor. If the number of identical particles N were just 2, then we’d correct for this by dividing by 2 so that we don’t count both AB and BA as different states. For N=3, we’d instead divide by 6 to account for the permutations ABC, BCA, CAB, CBA, ACB, and BAC. In general, we have to divide by N! (for example 3! = 3 x 2 x 1 = 6). which is the total number of ways you can permute N things. So the actual formula for the number of possible states one chamber can be found in is (n^N)/N!. When you take the logarithm of this to find the entropy, it changes an expression that for large values of N scales like N log n into one that scales like N (as long as we keep the ratio of N/n fixed). This is what we want —the entropy of large (macroscopic) systems should always scale linearly with the system size N.
Let’s check and make sure that this way of counting resolves the Gibbs paradox. If we take our formula for the number of states in one chamber (n^N)/N! and double both n and N at the same time, we get ((2n)^(2N))/(2N)! = 2^(2N) x n^(2N)/(2N)!. This is the total number of states the system can be in after we’ve opened the door. Before we open the door, squaring the number of states for each half gives (n^N)²/(N!)² = n^(2N)/(N!)². This is different from the previous expression only by a factor of 2^(2N) x (N!)²/(2N)!. This may look like a big difference, but for large N this expression just simplifies down to √(πN) (via Sterling’s approximation for N!). Because the square root of N is negligible compared to enormous things like N! or n^N, taking the logarithm of it doesn’t add anything noticeable to the entropy. In other words, for macroscopic systems, it doesn’t matter whether you count the states in each half of a system and then multiply them, or if you count up all of the states in the entire system at once. There is only one well-defined number of states, and hence only one unique value of the entropy (given by the logarithm of it). In other words, if you have two separate chambers of ¹H gas, the entropy doesn’t increase when you open the door and let them mix. Nor does it decrease when you close it again.
What about for the case where the entropy should increase (where we start with ¹H in one chamber and ²H in the other)? In that case, because the two different types of atoms are distinguishable, the total number of states in the system is different before and after you open the door. Before you open the door, it is the same as above — n^(2N)/(N!)². But after you open the door, we can’t just divide by (2N)! since there are not 2N identical particles. Instead, we have N particles which are identical to each other (the green ones), and N other particles which are identical to each other (the red ones). So the permutation factor to divide by is instead (N!)x(N!) = (N!)². This means the total number of states after the door has been opened and the gasses have fully mixed is (2n)^(2N)/(N!)² = 2^(2N) x n^(2N)/(N!)². This differs from the closed-door state count by a factor of 2^(2N). Which recovers our original conclusion (if we had ignored the permutation factors entirely) that there are 2N bits of information which become hidden after we allow the two different gasses to mix. This information can be thought of as the information about which side each atom originated on. It was clear where all the green ones were and where all the red ones were originally — this was part of the macroscopic state specification. But after the mixing it is no longer clear, everything is scrambled and the information has become lost in the microscopic details of the system. It hasn’t disappeared, but it has become hidden to the extent that it is hopeless to expect one could retrieve any substantial portion of it in any conceivable lab experiment.
If all particles were distinguishable, then it’s hard to imagine how thermodynamics as we know it could work. Without the indistinguishability of particles, entropy would not scale with the size of a system (even for large macroscopic systems) and there would be multiple different ways of computing the same entropy. This was the first hint that particles at some level must be indistinguishable. But since there are still multiple kinds of particles, it left the door open that there might be different species of electrons, or different species of ¹H atoms, just as there are different isotopes of Hydrogen atoms (¹H, ²H, etc.).
Gibbs had figured out the resolution to his own paradox already by the beginning of the 20th century — particles of the same type need to be treated as indistinguishable from each other in order for statistical mechanics to be able to reproduce all of the known results of thermodynamics and in order to give it a solid foundation. But it wasn’t long after that when quantum mechanics ended up confirming for other reasons that basic elementary particles like the electron must be multiple identical instances of the same thing.
I will leave that story for Part 2!
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