I am a bit puzzled by something (which I perhaps once knew quite well). I am
reviewing my basic quantum mechanics by watching Jim Branson's QM course on
streaming video. There is something he keeps say that bothers me.
He keeps saying that a wave function like exp(ikx) can't be normalized to
"one particle" and so must be a beam of particles?
....Huh?...
I understand what it means to say that exp(ikx) can't be normalized to one,
but that never meant to me anything about the number of particles to me. It
is just an idealized state that is not in technically in the Hilbert space
(we might use "rigged Hilbert spaces" for this kind of thing). So what is he
talking about? What am I missing?

But now this brings up an interesting question. In plain old quantum
mechanics (not QFT), the wave function for a pair of pairticles (moving in
1D for simplicity) is an L^2 function on R \times R. Thus it seems that
exp(ikx) can't refer to more than one particle in anycase! It is a
generalized eigenfunction of momentum for a single particle (isn't it?). The
wave function describing, say two particles must be a functions of two
variables like say \psi (x_1,x_2).
So wouldn't a beam of many particles have wave functions of many variable
(the number of particles)? And yet we hear that exp(ikx) is a beam of
particles!! Remember I am talking about a QM class here, not a QFT class (so
no Fock space etc).