I was recently teaching a grade 12 physics class and had to explain why only two electrons can be in the same orbital. My explanation went a bit like this (follow the links for explanations of the concepts):
Electrons are spin-1/2 particles. Particles with half-integer spin are called Fermions. Fermions have a really strange property: no two fermions can be in the same quantum state. Because of this, we can only have two electrons in the same energy level: one with its angular momentum pointing down (you can imagine it spinning clock-wise on itself) and one with its angular momentum pointing up (or spinning counter-clock-wise). After that, you’re out of options: the next electron has to be in a different energy level, since electrons are only allowed two possible directions of spin
The question that followed left me at a loss for words:
Why can’t two fermions occupy the same quantum state?
The physicists amongst you will have an answer ready: because of Pauli’s exclusion principle. For those of you who never heard about it, Pauli’s exclusion principle says that no two fermions can be in the same quantum state.
However, as an explanation this is not great. Basically what we are saying is: “no two fermions can be in the same quantum state because there is a principle that says that no two fermions can be in the same quantum state.”
Again, the physicists amongst you may have another answer ready: Pauli’s exclusion principle is, after all, a consequence of the spin-statistics theorem. What happens is that particles with a half-integer spin must have an antisymmetric wavefunction. This means that, if we exchange any two particles, the sign of the wavefunction must invert.
For those of you who are not physicists, this is harder to explain. The idea is that the probability of finding a certain particle in a certain state is given by this mathematical monster called the wavefunction. We calculate probabilities by taking its square (not exactly, but close enough.) A wavefunction can describe not only one particle, but several. In fact, in quantum field theory we don’t talk about particles but fields and the particle number can oscillate. When our wavefunction is antisymmetrical, it means that, by exchanging any two particles, we get a minus sign. That is:
Wavefunction (electron 1 in state 1, electron 2 in state 2) = -Wavefunction (electron 1 in state 2, electron 2 in state 1)
Now, what happens if two electrons are in the same state? Then we have:
Wavefunction (electron 1 in state 1, electron 2 in state 1) = – Wavefunction (electron 1 in state 1, electron 2 in state 1)
However, electrons are indistinguishable particles, so electron 1 = electron 2. Therefore, we have:
Wavefunction (electron 1 in state 1, electron 1 in state 1) = – Wavefunction (electron 1 in state 1, electron 1 in state 1)
There is only one number that’s equal to its inverse: zero! This means that the wave function for two fermions in the same state has to be zero. Since the wave function is the square root of the probability, we have that the probability of finding two fermions in the same state is zero. Therefore, no two fermions can be in the same state.
However, we still haven’t explained anything. Because the next question is:
OK, but why does the wavefunction of a fermion have to be antisymmectric?
And here I have to say I’m stumped. Yes, I know how to derive this mathematically from Dirac’s equation, but I have no idea how to explain it in any mildly intuitive way. I have also been looking online for an easy-to-understand explanation and found absolutely nothing.
Hence, I want to propose a challenge for my physicist readers: can you come up with an intuitive, math-free way of explaining Pauli’s exclusion principle?
I can’t offer a prize, but I can offer my undying gratitude and a link to you in my next (and this) article.
Also, you will have contributed to enlightening a very curious 18-year-old.
That’s gotta be worth something.
Here’s a new take on the issue from my new blog.