Re: uncertainty, wave function

In article <1130959507.247709.24850@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<gmarkowsky@xxxxxxxxx> wrote:
>Hi all,
>
>I have two questions:
>
>1. Can one express the uncertainty principle as a purely mathematical
>statement about Fourier transforms? Like, no mention of physics at all?

Any wave mechanics has uncertainty relations. For instance, the width in
frequency space of the sharp knock made by hitting two rocks together.
The shorter the duration in time, the wider in frequency it must be. The
Fourier transform only comes into it because you transform from position
space to momentum space in Cartesian coordinates with a Fourier transform.

More generally, for any observables P and Q,

delta p * delta q = -i/2 \int psi* [P,Q] psi dV

where [P,Q] is the commutator. P and Q are called observables, but in the
abtract they're anything.

>
>2. I was reading about a particle trapped in a box, so we look at it in
>one dimension, and the wave function has to be zero at the edges of the
>box, and this gives us a countable set of possibilities for the wave
>function, which gives us the discrete set of possible energies for the
>particle. The same thing, done in a sphere of radius r, gives us what
>as the wave function? And what possible energies?

For the particle in the box, energy eigenstates are plane waves. For the
particle in the spherical box, energy eigenstates are given by spherical
harmonics and spherical Bessel functions. Those functions have an
orthogonality relation similar to that of the sines and cosines of a
Fourier transform, and the transition from position space to angular plus
radial momentum space is done in a similar way.


--
"Voice or no voice, the people can always be brought to the bidding of
the leaders. This is easy. All you have to do is to tell them they
are being attacked, and denounce the pacifists for lack of patriotism
and exposing the country to danger." -- Hermann Goering
.

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