# FOM: Alice, Carol and Leibniz

Miguel A. Lerma mlerma at math.northwestern.edu
Wed Apr 17 11:36:51 EDT 2002

```[Matt Insall]
> [...] Would you say that any two
> electrons are indistinguishable, no matter what is their degree of
> separation, and no matter what states they are in?  It seems that protons
> are distinguishable, [...]

Indistinguishable or identical particles are particles with exactly
the same intrinsic physical properties: same mass, electric charge,
total spin, etc.  Any two electrons are identical.  Protons are not
elementary particles (they are made up of quarks and gluons), but for
many purposes they also can be considered identical (they behave like
fermions).  Of course two identical particles can be in different
states, which may allow us to "track" them separately, but only up to
some extent.  For instance, two electrons spatially confined in two
non overlapping boxes A and B have a zero probability of exchanging
their identities, so at any time we know with a 100% probability that
the electron in box A is "the same" electron it was in that box
before.  But if their wave functions overlap, then we partially lose
our ability to track them individually, and they may have a non zero
probability of exchanging their identities, so that if at a given time
electron 1 is in region A and electron 2 is in region B, later all we
can say is that electron 1 is in region A and electron 2 is in region
B with probability p, while electron 2 is in region A and electron 1
is in region B with probability q = 1-p.

These considerations are particularly important in studying
interactions among particles.  After a collision between two electrons
it is in general impossible to relate the identities of the electrons
that emerge from the collision with the ones that entered it.  An
analogy could be two waves traveling on the surface of water.  Up to
some extent we can talk of wave A and wave B as two separate
identifiable entities as long as they are far apart, but if they
collide and from the collision other two waves A' and B' emerge, the
question of whether wave A' is the same as wave A pretty much loses
its meaning.

Miguel A. Lerma

```