I did, and it states that electrons can hold one of two spin values. But within the mathematical framework of QM, that value is unknown before measurement - so it is ambiguous - which is interpreted (by QM) as a possible superposition of both values.
The physical experiment itself only detected spin up or spin down.
And that is the main question, how can we know this [electron superposition state] is not just a product of the framework being used, with no real counterpart in reality?
The real evidence of this is from the last experiment under "Sequential Experiments". Essentially, you only select particles with spin-up in the z direction, then you select particles with spin-up in the x direction. So the resulting particles should be up in both the z and x direction. But if you measure in the z direction again, you find an equal distribution in up and down, indicating that you can't simply treat the spin as both pointed in the x and z direction, and that measuring in the x direction has scrambled the previously well-defined z direction.
This experiment shows that measuring the spin direction in different axes does not commute. Measuring in one direction scrambles the other, which is equivalent to saying measuring in x then z is not the same as measuring in z then x. This fact is inherently related to the notion of a superposition. If a particle's spin direction is well-defined in one measurement basis, it is not well-defined in another - meaning it is in a superposition state in that measurement basis.
You might ask - why can't I describe the system after measuring in the x-direction as just a random mixture of up and down in the z-direction? Physicists use something called a density matrix to describe systems that have both some degree of quantum superposition and classical randomness. One way to measure the degree to which some stream of particles is a random mixture or not is to interfere particles in that stream with each other.
In the Stern-Gerlach experiment, after measuring in the x-direction, if the information in the z-direction was actually simply randomly scrambled, the probability that any two particles from the stream are truly identical is 1/2. If the particles are all identically in a superposition state, then any two particles will always be identical.
You can actually test the indistinguishability of two particles by doing an interference experiment. One very nice example of this is this experiment:
https://arxiv.org/pdf/1312.7182.pdf
Two atoms were trapped next to each other using lasers. If these atoms have the same spin, they're indistinguishable. If they have different spin, then they are distinguishable, and won't interfere with each other. In fig. 3, you can see varying levels of interference depending on how well-aligned the spins are.
I am not quite sure what you mean. There is tons of experimental evidence for quantum mechanics.
Specifically for electron spin, you seem to agree that it has spin 1/2 (because you say that it can be spin up or spin down). But that already concedes that there are superpositions of up and down. Spin left, spin right, spin straight forward, spin backward are all equal superpositions of spin up and spin down; and all these states are trivially constructed.
Bound electrons (in an orbital) can only have 1 of 2 quantum "spin" states - which is in relation to the nucleus and/or the magnetic field. Up or down. With the spin-up electron having a slightly higher or lower energy level then the spin-down electron (depending on other factors).
I think the question is how you unambiguously measure spin-right or spin-left as existing. Most experiments report only a stream on spin-up and spin-down measurements, inferring the original superposition.
