# FOM: Alice, Carol and Leibniz

Miguel A. Lerma mlerma at math.northwestern.edu
Tue Apr 16 13:06:36 EDT 2002

> <Later Miguel A. Lerma wrote:>
>
> >depending on how much their wave functions
> >overlap they always have some probability of exchanging their
> >identities, so you lose track of which one is which one - it is
> >not that we are "unable" to determine their individual identities,
> >if Quantum Mechanics is right, they do not even have a well
> >defined individual identity.
>
<wiman lucas raymond>
> I'm not sure that's quite right.  Each electron is defined in quantum
> mechanics by a certain kind of wave function.  A superposition of two
> (obviously normalized) wave functions is a linear combination of them
> that is also normalized.  In this sense, they do have distinct identies
> inasmuch as the decomposition into "elementary" wave functions is
> unique.

A linear superposition of two wave functions does not
represent the state of two particles, but a combination
of two states of the same particle - which is by itself
another state of that particle (think of an electron going
through two slits A and B as a combination of the electron
going through slit A and the same electron going through
slit B).  The state of two (distinguishable) particles is
the _product_ of their wave functions. In general the
space of states of a system made up of two or more
subsystems is the tensor product of the spaces of states
of the subsystems. But if the particles are indistinguishable,
then the state of the total system is not just the product
of the states of the individual particles, but the symmetrized
or antisymmetrized product of their states (depending on the
statistics, symmetrized for bosons, antisymmetrized for fermions)
So if we have two electrons, one in state |1> and another one
in state |2>, the state of the total system of two electrons
is represented by the antisymmetric product |1>|2> - |2>|1>
(electrons are fermions) - times the appropriate normalizing
constant. So it is impossible to tell which one is in state
|1> and which one is in state |2>, in a sense the question
is meaningless, there is just "some" electron in state |1>,
"some" electron in state |2>, and there are two electrons in
total. At any time there is even a probability of exchange, that
is, that the electron 1 becomes electron 2 and viceversa, given
by the probability of a transition of the form |1>|2> -> |2>|1>,
which is, I think, p = |<1|2><2|1>|^2,

Anyway, it has been some time since I studied this stuff.
Maybe someone else can provide more details about the way
elementary particles behave and whether they provide some
sort of counterexample to our intuitions on the foundations
of the concept of "number".

Miguel A. Lerma