Indistinguishability

In thinking about presenting physics to a lay audience, I think we haven't sufficiently emphasized what we mean by particles or objects being "indistinguishable" and everything that touches in modern physics.

The vernacular meaning of "indistinguishable" is clear.  Two objects are indistinguishable if, when examined, they are identical in every way - there is no measurement or test you could do that would allow you to tell the difference between them.  Large objects can only approach this.  Two identical cue balls made at a billiard ball factory might be extremely similar (white, spherical, smooth, very close to the same diameter), but with sufficient tools you could find differences that would allow you to label the balls and tell them apart.  In the limit of small systems, though, as far as we know it is possible to have objects that are truly indistinguishable.  Two hydrogen atoms, for example, are believed to be truly indistinguishable, with exactly the same mass, exactly the same optical and magnetic properties, etc.  

This idea of indistinguishable particles has profound consequences.  In statistical mechanics, entropy is commonly given by \(k_{\mathrm{B}}\ln \Omega\), where \(k_{\mathrm{B}}\) is Boltzmann's constant, and \(\Omega\) is the number of microscopic ways to arrange a system.  If particles are indistinguishable, this greatly affects our counting of configurations.  (A classical example of this issue involves mixing entropy and the Gibbs Paradox.)  

Indistinguishability and quantum processes deeply bothered the founders of quantum theory.  For example, take a bunch of hydrogen atoms all in the 2p excited state.  Any measurement you could do on those atoms would show them to have exactly the same properties, without any way to distinguish the first atom from, say, the fifth atom selected.  Left alone, each of those atoms will decay to the 1s ground state and spit out a photon, but they will do so in a random order at random times, as far as we can tell.  We can talk about the average lifetime of the 2p state, but there doesn't seem to be any way to identify some internal clock that tells each excited atom when to decay, even though the atoms are indistinguishable. 

It gets more unsettling.  Any two electrons are indistinguishable.  So, "common sense" says that swapping any two electrons should get you back to a state that is the same as the initial situation.  However, electrons are fermions and follow Fermi-Dirac statistics.  When swapping any two electrons, the quantum mechanical state, the mathematical object describing the whole system, has to pick up a factor of -1.   Even weirder, there can be interacting many-body situations when swapping nominally indistinguishable particles takes the system to an entirely different state (non-Abelian anyons).  The indistinguishability of electrons has prompted radical ideas in the past, like Wheeler suggesting that there really is only one electron.

TL/DR:  Indistinguishability of particles is intuitively weird, especially in the context of quantum mechanics.