HomesliceMMA said:
Man, this is so weird to me (and almost certainly so far above me). I ask anyone that can help - can ANYONE explain this in simple terms to me? And everything I've read (behaves like wave before, particle after, measurement) is dead wrong? I mean, I could google it now and get hundreds or thousands of people saying that same thing effectively. They are all wrong?
The confusion comes from the fact that unfortunately many popular-science book writers or, even worse, youtubers like to have quantum theory to present "something weird". They think they'd make the subject more interesting to sell their stuff better or getting higher click numbers. However, the really exciting aspect of science is that it describes the objectively observable phenomena of Nature in ever more clear and complete mathematical (!) models and theories.
All the "quantum weirdness" goes away, when you are accepting that the currently valid version of this theory is the one discovered in 1925-1926 in three equivalent forms: Born, Jordan, Heisenberg -> "matrix mechanics" (including the quantization of the electromagnetic field!), Schrödinger -> "wave mechanics", and Dirac -> "transformation theory". The latter is the most general scheme, which is based exclusively on the idea that the observables are described as a algebra of so-called self-adjoint operators on a Hilbert space, enabling the description of the symmetry principles already known from classical physics.
This mathematical formalism allows you describe the probability for the outcomes of measurements on a quantum system, which has been prepared in some way described by the quantum state. That's all that can be described by quantum theory (QT), and it's, as far as we know today, also the only description with is consistent with all observations ever made in attempts to testing the theory.
The confusing ideas of "wave-particle dualism" etc. is due to an old predecessor description of the behavior of nature in the quantum realm, rightfully dubbed "the old quantum theory". It was a collection of guesses, based on classical physics, adding some ad-hoc "quantum rules". This only leads to an apparent success for discribing the most simple systems. In fact it only works for the free particle, the hydrogen atom, and the harmonic oscillator (including the free electromagnetic field and thus also Planck's black-body radiation law). Today we know that's just, because these systems are described by equations of motion that have an exceptionally high symmetry, and in this sense that's just by sheer luck. It doesn't work already for the next-most simple atom, the Helium atom.
On top it leads to obviously wrong qualitative conclusions: E.g., according to the Bohr-Sommerfeld model of the hydrogen atom, you'd expect that such an atom is geometrically seen as a little disk, i.e., the electron runs around the proton in a planar circular or elliptic orbit (similar to the planets running around the Sun according to Kepler's laws). In fact a hydrogen atom in its ground state is a spherical object, and that's what's indeed observed in scattering experiments.
Even worse, as you realize yourself, the picture the "old quantum mechanics" provides as a description of nature, is intrinsically contradictory. Wave-particle dualism is the most infamous example for this: Of course, it's intrinsically inconsistent to think that a particle like an electron is both a wave and a particle at the same time, and that's resolved by modern quantum theory by an admittedly quite abstract description of what's observable about an electron in real-world experiments. The solution of the apparent wave-particle paradox is the probabilistic meaning of the quantum state, i.e., that all you can know about an electron are the probabilities for the outcome of measurements of some observables (its position, momentum, magnetic moment, etc.) given the state this electron is "prepared" in before making these measurements. In the wave-mechanics formulation the most determined states (so-called "pure states") are described by Schrödinger's wave function, whose meaning is that its moduluse squared provides the probability distribution for its position and the probability for finding one of the components of the spin in direction of an applied magnetic field used to measure it (e.g., in a Stern-Gerlach experiment) in either spin up or spin down direction (##\sigma_z \in \{\hbar/2,-\hbar/2 \}##).
