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Double slit and collapse
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The double slit experiment, where a broad beam of particles passes 
a screen with two slits, is one of the most fundamental quantum 
experiments.

Standard wave function arguments for purely unitary quantum mechanics
predict (at best) that the effect of the screen is to turn a particle
in a pure state psi into a superposition of at least three terms, 
one each for being in one of the two beams (for sufficiently wide 
slits) or spherical waves (if the slits are narrow enough) 
passing the slit and a third (or more) for the particle being stuck 
somewhere on the screen.
 
This conclusion is arrived at as a simple consequence of linearity of 
the Schroedinger equation, together with natural assumptions of what 
happens for particles prepared in coherent states.

But it is generally believed - and assumed in _all_ discussions of
interference - that a double slit screen projects a particle with
incoming wave function psi with the correct Born probability to a
particle in a superposition of the two beams that pass the slits.

The challenge is to derive this from a quantum model of the situation,
without invoking explicit collapse anywhere in the derivation.
Before this cannot be done convincingly, I don't consider the
measurement problem solved. 

For a precise version of a (slightly different) challenge, see
http://www.mat.univie.ac.at/~neum/collapse.html