Length of wavetrain of a single photon AGAIN!

[Ray Tomes]

Way back in the previous millennium, Ted Bunn wrote
a long explanation concerning the length of wavetrain
of a single photon and stored it on the internet at
http://www.lns.cornell.edu/spr/1999-02/msg0014640.html
where I just found it. Thank you Ted!

This got me thinking and I started messing about from
Delta x . Delta p <= h/2 to successively substitute
things for p like E and then f until I ended up with ...
Delta x . Delta f <= c/2 (yes, the h disappeared).
and I guess that if wavenumber is substituted for f then
it gets even simpler (I left this tiny step for the
dear reader to check that you are awake).

In this form, and armed with Ted's explanation, the
equation is not in the least mysterious. Rather, it
says that for a wave train of some length the number of
waves present is determined to an accuracy of 1/2
(or maybe its 1/2 a radian).

In such a form then it is a simple mathematical
statement about Fourier analysis which is not in the
least bit surprising. Is this simplicity understood
by quantum mechanics? Doesn't this mean that from such
a simply obvious mathematical fact it is then an easy
task to go backwards through the steps and derive
Heisenberg's uncertainty principle?

Why didn't they explain it this way in school?

Thanks again Ted Bunn.

--
Ray Tomes
http://ray.tomes.biz/
http://www.cyclesresearchinstitute.org/

[Blackbird]

Ray Tomes wrote:
> [...]
>
> In such a form then it is a simple mathematical
> statement about Fourier analysis which is not in the
> least bit surprising. Is this simplicity understood
> by quantum mechanics? Doesn't this mean that from such
> a simply obvious mathematical fact it is then an easy
> task to go backwards through the steps and derive
> Heisenberg's uncertainty principle?
>
> Why didn't they explain it this way in school?

I obviously don't know what school you went to, but this is certainly
understood. E.g., this is how Bohm introduces the uncertainty principle in
"Quantum Theory", 1951.

[Jon Bell]

In article <e5gigs$kou$1@lust.ihug.co.nz>,
Ray Tomes <ray@tomes.remove.biz> wrote:
[snip]
>In such a form then it is a simple mathematical
>statement about Fourier analysis which is not in the
>least bit surprising. Is this simplicity understood
>by quantum mechanics? Doesn't this mean that from such
>a simply obvious mathematical fact it is then an easy
>task to go backwards through the steps and derive
>Heisenberg's uncertainty principle?
>
>Why didn't they explain it this way in school?

Maybe where *you* went to school they didn't explain it that way. ;-)

Both of the textbooks that I've used in my sophomore-level "Introduction
to Modern Physics" course arrive at the HUP by discussing what happens
when you add a bunch of waves together, with a spread delta_k of
wavenumbers, to form a wave packet of width delta_x. They both note that
for a particular shape of wave packet, the product delta_x * delta_k is a
constant, then quote without proof the result from Fourier analysis that
the smallest possible value for that product, for any shape wave packet,
is 1/2, which leads straight to the HUP.

To reinforce this, I show my students graphs that I obtained by adding
together about 20 waves, first a set with one value of delta_k, then a set
with a delta_k which is 1/2 the value for the first set. By measuring
directly off the graphs it's easy to verify that delta_x for the second
set is twice that for the first set.

--
Jon Bell <jtbell@presby.edu> Presbyterian College
Dept. of Physics and Computer Science Clinton, South Carolina USA