Well, actually the wave/particle duality is one of the better understood aspects of quantum mechanics; it really follows from what's call the formal axiomatic formulation of QM. There are six of these (or five or seven, depending on how they're formulated. First of all, all information about a physical system is encoded in something call a state vector, whose time evolution under ordinary nonrelativistic conditions is specified by something called the Schrödinger Equation, an abstract entity defined in an infinite-dimensioned space of complex numbers called a H(ilbert) space, which has certain very nice, mathematically tractable properties. Each 'dynamical variable' of classical mechanics corresponds to a particular mathematical operator defined on vectors (vectors being certain complex functions of real variables, functions with those nice properties) in this H-space. For each such operator O, there is a class of vectors (with those nice properties!
) ø-1, ø-2... such that Oø-n = c-nø-n with the c's (So called eigenvalues of ø-n) all real numbers (you know how for example the first derivative wrt x of the logarithmic base e, say eˆ(3x) winds up just multiplying eˆ3x by 3? So that means that eˆ3x is an eigenfunction of the operator (d/dx), since applying that operator to the function eˆ3x just multiplies this function by 3. The second of the axioms, or postulates, of the formal theory of QM states that every H-space operator corresponds to a dynamical variable of the system, and every eigenvalue c-n of an eigenvector ø-n is a measurable value for the dynamical variable corresponding to O. Now remember, the state vector is the only source of information about the properties of the state. It follows from the third axiom of QM, which is a bit too mathematically complex for this site's text display capabilities, that only when the state vector coincides with some eigenvector ø-j of a particular operator corresponding to some variable (e.g. position, momentum, angular momentum, energy, etc.) will there be a probability of 1 for measurement of the eigenvalue c-j of ø-j. If the state vector does not coincide with ø-j, then c-j exists only as a possible value in a smear of probabilities. That's the mathematical background, and it's actually pretty clean. But look at what follows: assume that ø-j is an eigenvector of position. If the state vector happens to coincide with ø-j, then the nature of the operator corresponding to the position operator ˆP, which takes this eigenvector as its function, turns out to yield a real number as the eigenvalue, corresponding to a defined position in space with the value c-j. But if the state vector coincides with an eigenvector of the momentum operator ˆM, then the eigenvalue will turn out—because of what the eigenvectors of the momentum operators have as their mathematical form—to display values corresponding to a periodic function, i.e., a wave, because that is what the momentum eigenvector functions look like mathematically. Thus, the wave/particle 'duality' is nothing more than the fact that the form that the wave function takes is intrinsically indeterminate and unconstrained: under certain conditions it can be forced by the measurement setup to coincide with the eigenvectors of ˆP and under others it can be forced to coincide with ˆM.
When this rough sketch is fleshed out and made explicit and rigorous, it really corresonds to an extension of ordinary language, and we can indeed talk about wave/particle duality rather easily. That doesn't mean we can
understand it, but that problem isn't restricted to the very small quantum range; geneticists used to be able to talk about combinatory genetics without understanding what was going on, because Mendelian genetics was worked out long before the molecular structure of the macromolecule chains, DNA and RNA, whose replication is the basis for the transmission of biological properties over the generations, was known.
What's really needed are predictions that don't require access to energy regimes comparable to those in the Big Bang or very early universe, or even supernova conditions. Something that the LHC would be able to see, say, would do nicely....