It's been a while since I last discussed Quantum Mechanics. Let's do some more!
One of the aspects of Quantum Mechanics is the existence of identical particles. Suppose I'm holding a photon in my right hand, and a photon in my left hand. I can't physically hold a photon in my hand, of course, but consider it as an abstract example that I can illustrate with pretty pictures.
These two photons are identical. They are exactly alike in every respect. And I do mean every respect. Furthermore, this has profound physical consequences. By profound, I mean that you wouldn't be here if it weren't for identical particles.
Now, I could start talking about the mystical oneness and sameness discovered by scientists in the 20th century, but that would be quite meaningless. There is in fact some meaning I intend to convey here. I say that the photons are "identical" because if you switch photon A and photon B, nothing will change.
But as shown above, it sure seems like something does change when we switch the photons. Namely, before the switch, photon A is in my left hand, and photon B is in my right hand. After the switch, photon A is in my right hand, and photon B is in my left hand. Arguably, this is a very similar situation, but I said that the particles are identical in every respect, identical in a deep quantum mechanical sense.
Therefore, it is not the case that photon A is in my left hand while photon B is in my right hand. Instead, we have a mixed quantum state, with both photons in both hands. Being in a mixed state means adding or subtracting the wavefunctions of two or more states.
Recall that addition is commutative (ie x+y = y+x). And that's how switching the two photons leaves us with the exact same quantum state that we started with.
Okay, so that's nice, you say, but what physical consequence does that have? In this case, there is no consequence whatsoever, because the photons in my two hands are too far apart. But imagine that the photons were very close together, so that their wavefunctions were overlapping. Then we'd have some constructive interference in the overlap.
More impressively, some kinds of particles will have destructive interference. Let's say that instead of photons, we had electrons in our two hands. The two electrons are identical in every respect. But unlike photons, if we switch two electrons, one thing does change. That is, after switching, the wavefunction will be the negative of what it was before. Therefore, the quantum state of the two electrons will have subtraction instead of addition.
Aside: I am representing photons with wavy things, and electrons with fuzzy circles, but the intelligent reader will realize that these are just artistic representations with little basis in reality.
Now imagine that the two electrons weren't in two different hands, but were both in the exact same spot, with the exact same momentum and spin and everything. Then the wavefunction would be something like this:
But I just took a wavefunction and subtracted it from itself. That's just zero! A wavefunction of zero is not possible. Therefore, it is not possible for the two electrons to occupy the same state.
Consequences? To start, this is why atoms have electrons in a complex orbital structure. If the electrons weren't identical, then they would all fall to the lowest orbital around the nucleus. Chemistry would be very different, and you can just forget about biology. But because they're identical, electrons cannot all occupy into the same state. Instead, electrons are forced into higher energy states around the atom, with the highest energy electrons doing all the chemical reactions.
A more astronomical consequence: White dwarfs. White dwarfs are small stars that are held up by so-called electron degeneracy pressure. The electrons can't compress into a smaller volume, because that would require them to fill up the higher energy states. The gravitational force is not enough to supply this energy. However, if the star accumulates enough mass, perhaps from a nearby star, it reaches the Chandrasekhar limit, where the gravitational force is strong enough to supply the extra energy. Upon reaching the Chandrasekhar limit, the white dwarf will collapse, causing a supernova.
Neutron stars are very similar, except they're held up by neutron degeneracy pressure.
Neutrons, like electrons, will interfere destructively with each other. Particles which interfere destructively (like electrons) are called fermions. Particles which interfere constructively (like photons) are called bosons. Fermions include electrons, neutrons and protons. Bosons include photons, gravitons, and certain atomic nuclei, like Helium-4.
Identical particles are all around us. They exhibit weird and important properties that have no analogue in classical mechanics. This is why quantum mechanics matters.