I read this stuff in my spare time (because I'm fascinated by it!) and have no actual physics background, so please bear with me as I try to ask this question that has been bothering me:

Right now I'm reading a book by Brian Greene called 'The Fabric of the Cosmos'. In his chapter on 'Entangling Space' he describes the probability wave - which I've heard the term before - but am only now seeing what it is. This quote (paraphrased) from the book is really confusing me:

That last line really throws me. We know that a hydrogen atom has 1 electron, and we can't KNOW it's position before we measure it. So if that's the case how do we KNOW that it is in an orbit around the proton unless we actually look? It looks like understanding probability waves goes a long way to understanding QM, so I'd like to try and understand this as best as possible and I'd very much appreciate any assistance.

This is one of the key features of QM.We don't know,unless we measure.Even if we measure,all we do know is just a probability/probability density.We assume the electron is somewhere around the nucleus and all we can do is tell with what probability we find it in a certain space volume,not necessarily near the nucleus.

This is the core idea of quantum mechanics. The only question is whether the fixation of the position (and of its other properties) occurs only when someone observes it, which brings human consciousness into the question, or when it interacts with other particles, which permit quantum physics to take place in places where there are no humans, like the core of the sun. Mathematically an "operator", a mathematical object similar to a matrix, acts on the wave function (which is also like a vector) and produces the probabilities of what will be observed (or take place) in our spacetime.

We can know the electron is bound to the atom by many many observations and by the consistent theory which has been formed (Shroedinger's equation for non-relativistic calculations and quantum field theory for relativistic ones), These theories are fantatsically accurate in predicting the values of measured constants and physicists have great confidence in them. They predict the various energy levels and shells and occupancies that electrons possess in an atom, and indeed the application of this knowledge revolutionized chemistry.

But the bottom line is that most of the time the electron is not localized in our spacetime, and our only description of it is this wave function, expressed in complex numbers, which only when acted on produces the probabilities that Greene spoke of. The rest of the time it evolves smoothly but does not change its nature.

I'm beginning to see why there are so few people fascinated in QM! So few people seem to 'get it'. Unfortunately I AM fascinated by QM, and more unfortunate I can't really take the time out to go back to school and get caught up on the basics. So that leaves me with books, the Internet and forums like this one.

So if I'm understanding this, because we can't actually see inside an atom (that would be nice!) then we have used theory and math to determine the atom's model - this is how we have determined the number of electron shells in each atom? We know the number of electrons allowed per shell but unless measured we don't know WHERE in the (s)hell a particular electron is?

That's pretty much the story.Heisenberg's HUP won't allow us to determine accurately the position of the electron inside the atom,as i said,all we can do it statistics.