Young's two-slit spatial probability density (Feynman Lectures vol 3.)
> >
> > I12 = I1 + I2 + 2*sqrt(I1*I2)* cos(theta) (1)
where I1, I2, I12 are intensities.
> > which is modelling to the probability equation:
> >
> > P(A or B) = P(A) + P(B) - P(A and B) (2)
> >
> > the interference term of (1) is simply the dot product of the
> > amplitudes |A|*|B|*cos(theta) and is therefore a measure of
> > their degree of orthogonality.
>
> Except that you don't normally get a negative number for P(A and B)
> in probability theory, particularly in the kind used in fuzzy logic.
> That is why QM is an _exotic_ probability theory, and why you cannot
> un-mix a superposition as Mati and others correctly point out.
Probability theory is not without it's deeper aspects:
http://www.seanet.com/~ksbrown/kmath309.htm
There is no reason confine probabilities to positive numbers and
Feynman (~1987) has already proposed using negative numbers.
The reference above shows how negative probabilities bring on
a nicer symmetry and how complex probabilities have some
really interesting properties.
The interference term is not present (theta=90) when complete
knowledge of which path the particle took through the two-slits
is given (Maxwell-Boltzmann)
When no knowledge of the paths is available, the particles are
called "identical" (theta=0) and the set overlap is complete.
But this is a macroscopic discription of identicality. (Bose-Einstein)
The particles were not identically prepared and still have a
variation in their momenta which is spatially dispersed by the slits.
When this variation is not present you have a Bose-Einstein condensate.
When partial knowledge of the paths is available, (Feynman
top of pg 1-9) some interference is present and is represented
by a set intersection in which the degree of set intersection
is given by the phase difference theta. (that's why you take the
absolute value of the amplitudes, theta is not free outside 0-90
in |A|*|B|*cos(theta) )
http://www.bestweb.net/~ca314159/SETPHASE.GIF
When intensities are used instead of amplitudes, the intensities
are given the frequentist interpretation in terms of the number
of particles. There is no superposition in this case. But theta
here is called by Feynman, the phase differnce, and it is not
clear whether he still confines it to 0-90 as is implied in the
dot product.
http://www.bestweb.net/~ca314159/INDEX.HTM
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