Charles Francis wrote:
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.
A wave function describes a single-particle state in the case of
fermions.
In the case of photons, the state is a product of the mode function
and the wave function, and the wave function depends on the number
of photons. It has to, because the mode function knows nothing about
photon number, and the wave function is the only place such information
can be stored.
The photon case is a generalised version (in a sense) of the quantised
simple harmonic oscillator, and even a brief read over that subject
in a sutable text clearly shows that the allowed wave functions are the
Hermite polynomials H_n, and that H_n is the wavefunction for an n-excitation
state of the QSHO.
The reason "a wave function like exp(ikx) can't be normalized to
one particle" is that the integral of its norm over all space is
not finite. Thus either we (a) box-normalise, and say there is one
particle per box (of sides L), and hence have some sort of beam;
or we (b) restrict the domain (e.g. to between -L/2 to +L/2) so the
the integral is finite, and hence normalizeable (although then
exp(ikx) is unlikely to be a good wavefn).
--
---------------------------------+---------------------------------
Dr. Paul Kinsler
Blackett Laboratory (QOLS) (ph) +44-20-759-47520 (fax) 47714
Imperial College London,
SW7 2BW, United Kingdom. http://www.qols.ph.ic.ac.uk/~kinsle/