Pages 20 and 21 of this article, http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf, appear to prove that electrons in the double slit experiment must be thought of as actually being in the superposition of two states at the same time. He uses the interference term in the probability distribution to justify his proof. I would be very much interested if someone(s) with more knowledge than me can read pages 20 and 21 and provide a technical review. (I have read other articles where it is stated that the interference in the double slit experiment comes from the fact that the electrons travel different path lengths when going through the different slits caused by the differing distance the electron is from each slit)

They are in a superposition of an innumerable number of states in many many different ways at the same time. Do you actually know what a vector space is? How many ways can real numbers be summed to make 1? And that decomposition is all at the same time. Its exactly the same with superposition. In fact real numbers obey the axioms of a vector space which you should look up.

They are in a superposition in the same way that a die is in a superposition when you shaking it before the role. It is a mathematical abstraction that allows us to calculate the probabilities correctly. It is the way you calculate probabilities in n-dimensional vector space.

For each different orthonormal basis there is a corresponding decomposition of the probability amplitudes. I assume this is what you mean by they are in a superposition in many different ways. But this also tells you that this superposition is not an actual physical state of the particle.

The superposition state cannot be one of the real states of an observable. Why? Because it is not a possible outcome. There are only n possible outcomes given by the n eigenvalues for the operator matrix. If you consider a superposition state to also be an allowed outcome then the number of possible outcomes increases, infact the number of possible outcomes would be infinite for every obsevable.

Yes I know what is a vector space. Let me ask you a question. When you are shaking 1 six side die, do you consider the die to be in a superposition of all six possibilities?

This is where the classical picture breaks down. A quantum system in an eigenstate of ##\hat{S}_x## is in a superposition of eigenstates of ##\hat{S}_z##.

I am not sure that you understand what is a vector space. Frankly, I do not see why you are making such a big issue of this. I do not think that you are thinking about this correctly. Your answer suggests to me that you support the proposition that a particle can be in two, mutually exclusive states at the same time. I would like you to explain, simply why you believe that.

I think what we are being told is that the same physical state may be described using different 'coordinates' - ie a change of basis. This requires that the state vector is in a vector space. So a state z-spin = 1 ( setting ##\hbar=1##) can also be described as a 'superposition' of thermal x and y spins.

Hence, I don't think any spin property has two values at once.

Regarding interference, you are right to be skeptical about the role of superposition and examine each case.

Thousands of experiments lead to this conclusion. It is not a matter of belief, it is a scientific result.

You asked for help from people with more knowledge. You got replies from experts. Now you question the knowledge of these experts - and not even for something advanced, but very elementary mathematics?