Conference rusure::math

A monochromatic beam of light is composed of particles (photons) each of which
has energy E=h*nu where nu is the frequency of light.  The more intense the
beam, the more photons you have.

A single photon has associated with it a probability density function
Phi(x,y,z,t) which describes the probability of finding the photon at
position and time (x,y,z,t).  (The probability density is Phi times its
complex conjugate.)  You can Fourier-decompose Phi into pure frequencies.
If the photon's position is known exactly then this decomposition cotains
all frequencies.  

The important thing is not to confuse these frequencies with the frequency 
of the light -- they are distinct quantities.

� Physics tells us that as the frequency of propagating electromagnetic energy
� gets higher, the energy behaves more and more like a particle.

  Actually, there is a duality between waves and particles, for both
  massless particles (photons) and other particles like electrons.  Whether
  something behaves like a wave or particle really depends on what your
  observables are, and Heisenberg's uncertainty principle states that
  you cannot observe both position and momentum with arbitrary accuracy
  simultaneously.  The position and momentum operators do not commute.
  This is mathematically a relation between the rate a time domain signal
  and its bandwidth decay in their respective domains.

� It also says that the energy propagates in packets that have total(?)
� energies that are multiples of h*nu where nu is some frequency associated
� with the particle.

  When an atom drops to a lower eigenstate, it emits a quanta of energy
  as a photon; the quantum satisfies Schroedinger's wave equation much like
  the way a harmonic oscillation will satisfy the boundry value problem
  of a vibrating string.  The atom acts as though it 'rings' for a few
  nanoseconds, and the frequency of the ringing is nu and is related to the
  change in energy as you stated.  But since the packet has finite duration
  it is effectively modulated by a Gaussian envelope and so really has
  a Gaussian frequency domain spectrum as well.  Depending on how you
  observe the photon it will look like an electromagnetic wave, or a
  particle, or some combination - but again, you cannot violate the
  uncertainty principle.

  I think the envelope of the 'packet' is really a side effect of how
  you observe it, but don't remember exactly.

  A really good place to read up on this would be the Feynman Lectures
  on Physics, especially volume 3 (on quantum mechanics).

  - Jim

  In looking at .1, the 'frequencies' mentioned in relation to the
  position are really the particle momentum; formally position and
  momentum play the role of time and frequency in the Fourier transforms
  that relate them.

  There is another 'frequency' also - in an atom the wave equation has
  spherical harmonics as solutions, which are an interesting example of
  harmonics existing over a non-commutative group (the group of orientations
  of a sphere).  The Fourier transforms one is used to are over the group
  of translations along a line, and are particularly simple as a result -
  but you can do all the usual stuff like convolution and transforations
  if you well-define your integrals over any continuous group, (the so-called
  Lie groups).

  - Jim

re .1:

>A monochromatic beam of light is composed of particles (photons) each of which
>has energy E=h*nu where nu is the frequency of light.

Does this mean that nu is a macroscopic parameter associated with the beam 
that bears no direct relation to the wave function of the particle ?
I'm still a bit confused.  What do we mean when we say 'a beam of light
has such-and-so a frequency' ?

>A single photon has associated with it a probability density function
>Phi(x,y,z,t) which describes the probability of finding the photon at
>position and time (x,y,z,t).  (The probability density is Phi times its
>complex conjugate.)  You can Fourier-decompose Phi into pure frequencies.
>If the photon's position is known exactly then this decomposition cotains
>all frequencies.

OK, but this is not nu, right?  Nu has to do with the macroscopic frequency,
right ?

>The important thing is not to confuse these frequencies with the frequency 
>of the light -- they are distinct quantities.

OK, but do they bear some relationship to one another ?  For example, is the
macroscopic frequency measured something like the the modulating frequency
for the photon packet ?  My model here is that the photon can be viewed
(at a single point in time) as something like a Gaussian envelope modulated
by some frequency (call it f0).  That modulating frequency (f0) would be
the macroscopic frequency we measure, even though the signal has Fourier
frequency components splashed around f0.

Did I get it right ?

re .2:

Aha! sorry about .4; I wrote it before I read .2.  This answers my question.
Thanks.