More a question to some conceptual understanding of combinatorics. The

[aaaa202]

More a question to some conceptual understanding of combinatorics. The number of ways of picking Na elements to be in a box A, Nb to be in a box B and Nc to be in box C is given by:

N!/(Na!Nb!Nc!)

One can proof this by saying: Suppose we start by putting Na in box A and so on. Now I have always wondered why on a deeper level you get the same result no matter on which box you start to put in your elements. Do all physical systems obey this kind of counting logic?

Edit: hmm.. maybe this is all gibberish..

 

[Ray Vickson]

More a question to some conceptual understanding of combinatorics. The number of ways of picking Na elements to be in a box A, Nb to be in a box B and Nc to be in box C is given by:

N!/(Na!Nb!Nc!)

One can proof this by saying: Suppose we start by putting Na in box A and so on. Now I have always wondered why on a deeper level you get the same result no matter on which box you start to put in your elements. Do all physical systems obey this kind of counting logic?

Edit: hmm.. maybe this is all gibberish..

Actually, this is not gibberish, but, rather, an important question. The short answer is: classical objects obey the counting above (called Maxwell-Boltzmann statistics), but atomic-scale (quantum) objects might not. In fact, for integer-spin objects (Bosons) the formula for the number of ways of packing N identical objects into three "boxes" would be very different, while for half-integer-spin objects (Fermions) it would be impossible to pack more than one object into each "box" (depending on exactly what you mean by a 'box'). See http://en.wikipedia.org/wiki/Bose–Einstein_statistics for a more detailed discussion, and see the links therein to Maxwell-Boltzmann and Fermi-Dirac statistics.

We know that in order to have results that agree with experiment, it is crucial to do the counting properly, because classical counting can lead to results very different from what we see in the lab.

Essentially, the classical approach assumes that while the objects are not distinguished, they are in principle distinguishable (so, for example, it makes sense to speak of object1, object2, etc., even if your counting formula _ignores_ the actual ordering). However, in the quantum world at the atomic or subatomic level, particles may be truly indistinguishable, so you cannot even speak of particle1, particle 2, particle3, etc.---you can just say you have three particles.

RGV