molecules - What actually is the Wavefunction?

I am aware that the square of the Wavefunction gives the probability density of finding an electron at a particular point in space. I have also heard that it's a complex number but since it's a function I am unsure as to how that could be the case (perhaps someone could please clear that up for me as a sub-question). Moreover, I have seen it described as the amplitude of something but I literally have no idea what is meant by that. However, the crux of the question is what is the Wavefunction itself (i.e not what the square of it is). I have read in the book "Why Chemical Reactions Happen" that it is essentially synonymous with the word/concept: orbital. I have also seen in various places w(x,y,x) or w(r,theta,thi) [where "w" represents the sign for a wavefunction] so, from what I can gather it is a function in three dimensions that represents the shape of a particular orbital where the function of (x,y,z) or (r,theta,thi) would, of course, be different for the s,p,d and f orbitals. However, in the book "Why Chemical Reactions Happen" it goes onto say that wavefunctions of different atoms interfere with each other to form molecular orbitals and the atomic orbitals somehow have a phase?! For my idea of a wavefunction to be correct, the idea of interference of these wavefunctions must be an idea that was thought of to try and rationalize the process of the formation of molecular bonding and anti-bonding orbitals from atomic ones to make it easier to understand and conceptualize. If I am right about the wavefunction being a purely mathematical representation of the space where an electron is likely to reside then the molecular wavefunctions are just as curious in terms of their shape and suchlike as the shapes of the atomic orbitals they came from (i.e nobody really knows why they are that shape but when atoms or molecules are arranged in a particular way, orbitals "just are" that shape). Also the idea of phase seemed to bring my argument down but then I thought that perhaps phase is simply defined as being opposite relative to another part of an orbital if it is the other side of an angular node (I suspect this is wrong but could someone please shed some light on this?)

Going back to when I said - "the molecular wavefunctions are just as curious in terms of their shape and suchlike as the shapes of the atomic orbitals they came from" - I thought this because I found it bizarre that empty orbitals could "interfere" with an orbital from another atom. For example one p orbital is empty yet all three interact with the four hydrogen s orbitals. Unless, the electrons can jump between the the degenerate p orbitals at such a frequency that all can be considered occupied I find it incomprehensible that just a portion of space can interfere with an orbital. As a result, I came to the conclusion that I quoted at the start of the paragraph. Basically meaning that when atoms come together to form molecules the electrons find the lowest possible energy spaces to reside; or in other words, a molecuar orbital / molecular wavefunction forms which is equally as curious as the atomic wavefunctions with their strange shapes.

Any help with this concept would be greatly appreciated - I am off to Oxford university this October and Molecular Orbital Theory is a small part of the suggested reading list before arrival into the first year. Also, on that note, please refrain from heavily mathematical answers because I simply won't understand as I have only done A-level Maths and A-level Chemistry. Thanks.

Answer

Let me see if I can get at some of your questions. As mentioned above, it's much easier when you ask individual specific questions.

One problem with books on introductory quantum mechanics is that, put simply, the language of quantum mechanics is math. Specifically, most people use the Schrödinger equation which involves second derivatives and differential equations.

When I teach quantum chemistry to undergraduates, there are typically problems due to the mathematical nature.

There are multiple "interpretations" of quantum mechanics. That gets into philosophy. Wikipedia has some nicely-written descriptions.

As you mention, the most widely used interpretation (the Copenhagen interpretation) of the wave function centers on the square of the wave function, or rather $|\psi^*\psi|$ as a probability density. As you mention, the wave function could be imaginary or complex, so this notation indicates a mathematical way of getting a real number for a probability density.

You ask about the shape and why atomic orbitals have particular shapes.

One thing that's amazing about quantum mechanics is that we put in some relatively simple equations to describe the motion of an electron in a hydrogen atom, and the solutions are exactly these atomic orbitals (s, p, d, f, etc.).

So I disagree that "no one knows" why atomic orbitals have particular shapes. The answer is that they are the mathematical solution to the Schrödinger equation. That is, we want to know the energies of the system and there are only certain quantized solutions.

And yes, the whole notion of chemistry is that electrons (and hence orbitals) on different atoms interact. Otherwise, there is no bonding.

I'll leave you with this. Without the math, it can be exceptionally hard to understand quantum mechanics. When you do understand the math, I think the results are quite beautiful. The shapes of orbitals aren't arbitrary. This is actually just a reflection of the mathematical nature of quantum mechanics.