one or many particles?

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Cyberkatru - 01 Jan 2006 20:39 GMT

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).

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Tao - 25 Feb 2006 11:15 GMT

>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
[quoted text clipped - 21 lines]
> particles!! Remember I am talking about a QM class here, not a QFT class
> (so no Fock space etc).

My understanding is that you are correct, and I have often observed the
same. Any multiple particle system should really have a wave function with
extra free variables for each of the particles in the system. Ignoring this
may give useful approximations in some cases, but is fundamentally
imprecise.

For example it is not precise to deal with an atom using wavefunctions for a
single electron and a nucleus, as there is certainly some interaction
between the electrons. Unfortunately dealing with the full equations is
difficult so "perturbative" methods are used as an approximation. The same
applies to a beam of electrons. If the beam was fairly dense, but of narrow
width it would clearly spread as a result of the electrostatic repulsion
between the individual electrons. A single electron does not "spread" with
this mechanism.

For a single electron, one can find an infinite sequence of wavefunctions
which are localised, but which eventually approximate a plane wave to any
required accuracy in any finite volume. The pointwise limit of this set of
functions is zero everywhere, but in distribution theory, the sequence
determines a distribution which gives zero probability of being in any
finite volume, but a probability 1 of being somewhere in the (assumed
infinite) whole space. The dual example to this one, where we work in phase
space (i.e. momentum space), finds a series of waverfunctions with
increasingly localised momentum. The limit of this set of wavefunctions
gives a Dirac "delta function" in phase space which is normalised but zero
except at a single point, indicating absolutely fixed momentum (the plane
wave is dual to the delta function, giving completely uncertain position ).
Dirac, of course created relativistic quantum mechanics as well and used the
delta function informally in the development. This viewpoint was given a
concrete mathematical foundation in distribution theory, invented by Sobolev
a few years after Dirac first invented the delta function for quantum
mechanics (the theory essentially constructs the dual to a space of
functions. Some elements of the dual are identified with normal functions,
but some are not, but are defined as the limits of certain sequences of
functions, as in the example above.

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Natural Science Forum / Physics / Particle Physics / February 2006

one or many particles?