Two Bosons in a Box: Is it 2/3?

[nonequilibrium]

Say we have a small box with two physically identical bosons in it. Is it true that if I partition the box in two equal parts, it's more likely (2/3 chance) I'll have both particles at the same side? (Note: classicaly there's "only" a 50% chance of them being together)

 

[Cthugha]

If you want a strict answer, this will depend somewhat on your initial conditions. However, if you are only interested in the easy picture, the probability to find the bosons in the same place will be twice as high for ideal bosons than for independent particles. This is called bunching.

This has been tested experimentally for example by dropping (non-ideal) bosons and checking whether their landing positions are independent or not (http://arxiv.org/abs/cond-mat/0612278" , also published in Nature 445 (2007) 402).

Another famous example is the Hong-Ou-Mandel experiment, where two indistinguishable photons entering a beam splitter at different entry ports will always exit via the same exit port.(http://en.wikipedia.org/wiki/Hong–Ou–Mandel_effect" ).

Also it might be worthwhile to note that this statistical tendency for bosons to arrive in pairs shows a factorial increase on the "order" of the pair. If you drop thousands of bosons and check their landing positions, you will find that boson pairs landing at the same position will appear twice as often as expected for independent particles, boson triplets landing at the same position will appear 6 times as often as expected for independent particles, boson quadruplets landing at the same position will appear 24 times as often as expected for independent particles and so on and so on.

 

[nonequilibrium]

amazing, thank you, that is exactly what I was looking for :)

I suppose there's no way to understand this classically (some texts explain it by postulating bosons are "truly indistinguishable" which then let's you use classical counting methods to get the same result, but that seems misguided?)

 

[Cthugha]

Well, the tendency for bosons to arrive in a correlated manner is basically reflected in the variance and higher order moments of the underlying probability distribution. Therefore you can identify that behavior for a large number of particles somewhat classically as you will get a Bose-Einstein distribution instead of a Poissonian one. However, in order to derive why a Bose-Einstein distribution develops, you need to treat this using quantum mechanics.

Basically, you will find out that - due to the different commutation relations - probability amplitudes for processes starting from two indistinguishable particles in some state and ending up with those being in the same state interfere constructively for bosons leading to this bunching tendency and will interfere destructively for fermions as is well known from the Pauli exclusion principle.