Thus spake Cyberkatru
OK, I am reviewing my basic quantum mechanics by watching Jim Branson's
QM class 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?

You have it right. A wave function describes a single particle state. If
he is really saying this, ignore him and find a teacher or a book that
understands the subject.

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).

In a beam such as a laser all particles may be in the same state. Then
you can use one wave function. Otherwise not.


Regards

--
Charles Francis
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