I was thinking about the nature of the atom, specifically, why electrons do not spiral into the nucleus. My physics book says the principal quantum number $n$ must be an integer number of wave patterns, four quantum numbers can never be in the same state (the exclusion principle) and angular momentum is quantized. But why? Is it simply because that is how quantum mechanics works?
Well, we could say, yes, that is simply how quantum mechanics works. But these are not the axioms of quantum mechanics, and the exclusion principle in particular is really only understood in the context of quantum field theory.
The electron does not "spiral in" because it doesn't move in the classical sense at all. At the scale of the size of an atom, quantum mechanics is inevitable, and classical reasoning is simply invalid.
That angular momentum and the principal quantum number are discretized follow from the analysis of the quantum mechanical operators associated to them (and is experimentally supported e.g. by the observation of discrete spectral lines): It is an axiom of quantum mechanics, the Born rule, that observables are only ever measured in the discrete spectra of operators associated to them, and it follow from general mathematical considerations (together with the axiom that the physical states are really well-represented by a separable Hilbert space) that the states belonging to these measurements - the eigenstates, labeled by their discrete values for the operator - are a basis of the space of states: All possible configurations of the system are only superpositions of these eigenstates. Analyzing the Hamiltonian (the energy) for an atom, one finds, best seen in the hydrogen atom, that such a basis is given by labelling the basis states by the four famous quantum numbers.
The exclusion principle is another beast entirely. It comes out of nowhere in ordinary quantum mechanics, and could be taken as another experimental/axiomatic input. However, in the more general framework of quantum field theory, we can derive the spin-statistic theorem telling us that fermions - the type of particles to which electrons also belong - never "share" a state - there is at most one particle in a given state.
At the end, you could now ask why the axioms are the way they are, why quantum field theory is the way it is. Well, there's no reason other than that it matches experiments well, and has hitherto been more successful in that than anything else.
Why is not really important, how is. If you ask yourself why then the answers can be many, for example
- Why does gravity make two masses attract each other?
The answer is because it does, what is really important is how and for that you have a first theory, Newton's Law of Gravitation, this theory is only true for relatively small masses (or masses with small density). There is another theory that works in any case, Eienstein's General Relativity.
Whether you apply one theory or the other, why the phenomenon happens should have the same answer in both cases, and it is in fact a philosophical answer. You can especulate like Fyenman does in The Relation of Mathematics & Physics, but it doesn't matter as much as how the system develops under the action of a gravitational field, a question which can actually be answered with the theories mentioned above.
Then, the fact is that the electrons do not orbit the nuclei like a planet orbits the sun, nor do they emit electromagnetic waves due the their centripetal acceleration, and yes they don't loose all their kinetic energy and collapse.
There is no need for a picture of how electrons move in their orbitals, it's enough to know that it's not like the classical physics would predict.