On Jul 12, 2009, at 12:53 PM, Abd ul-Rahman Lomax wrote:
I think the temperature is misleading. What matters is the
*relative* energies of the two molecules; if they happen to have
low relative energy -- the opposite of what we thought would be
needed! --, they are as if at very low temperature.
In other words, what we think of as a BEC, as a bulk phenomenon,
requires very low temperatures. However, if all we are considering
is two molecules, can those two condense in the same manner as a
BEC? I don't see why not, but, then again, I really don't know
enough to do more than ask a few questions.
I'm no expert in QM. I'm a complete amateur. However, I do know the
waveform of a BEC, even a two molecule BEC, can not be described as
simply the overlap of individual waveforms, and things do not behave
as one intuitively might expect. For simplicity lets just talk
particles instead of molecules for a moment. Yes, as relative motion
of any two particles is reduced to zero in the reference frame of the
observer, their de Broglie wavelengths increase to infinity, and
obviously greatly increase their overlap if the centers of charge are
not co-centered. This is not sufficient to increase the probability
of fusion. What is important is, upon observation and wave function
collapse, the probability of the two resulting point particles
(nuclei) being sufficiently close to produce the fusion. If you
break the individual wave functions into little cubes of a size
sufficiently small to produce fusion of two particles within one,
then as the wave function gets bigger you end up with more cubes
(i.e. proportional to the wavelength cubed number of cubes) even if
the wave functions *fully overlap*, i.e. the particles are co-entered.
If, for the sake of argument, each cube has equal probability, i.e.
upon wave function collapse each particle can be found in any of the
cubes with the same probability, then the probability of both
particles occupying the same cube upon full collapse actually
*diminishes* with expanded de Broglie wavelength. For example suppose
you start off with 2^3 = 8 cubes. The probability of fusion in any
one of the cubes is 1/8^2 = 1/64. You have 8 cubes, so the overall
probability is 8*64 = 1/8. Suppose now you double the wavelength, so
have 4^3 = 64 cubes. The probability of fusion in a particular cube
is 1/64^2 = 1/4096. The combined probability of fusion, given there
are 64 squares is 64 times larger, thus 1/64. The probability of
fusion is reduced by a factor of 8 when the de Broglie wavelength is
doubled (in this highly simplified version that is.)
It gets worse. The probability could in actuality in all low speed
cases be very close to zero. This is because the expected location
(upon wave function collapse) of particles in combined wave functions
is co-located with respect to the other particles, i.e. is co-
dependent. The probability of finding of particle A in a given cube
is conditional upon where particle B will be found, and vice versa.
Particles having like charge have low probabilities of being found
close together. This co-location affects things like the tendency
for hydrogen molecules to be of a given barbell shape and size. You
might expect that, as the protons are brought closer to each other,
and the volume of the molecule decreased, the probability of the
electrons being found between them and thus shielding their Coulomb
barriers, would grow. Not so. The electron wave function actually
thins out between the nuclei and thus increases the repulsion between
the nuclei, thus restoring the molecular shape. The probability of
the electrons jointly being found in the smaller volume between the
nuclei decreases, and the probability of both being found on opposed
sides of the nuclei increases. The probabilities are thus co-
dependent. Similar arguments can be made for nuclei jammed into a
tetrahedral space (their locations are co-dependent) as well as for
any electron screening that might occur there.
I suggested a possible means to beat this co-location problem (and
thus cause fusion) here in 1996. It is described here:
http://mtaonline.net/~hheffner/BoseHyp.pdf
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
