The "interpretation" of spin superposition is that a particle can be both spin "up" and spin "down" along some direction $ \vec r $ simultaneously. But what does this actually mean, since by ordinary vector addition, these two opposite vectors would cancel out to give a net result along of zero along $ \vec r $ ?
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Well, the answer is basically learning the formalism of Quantum Mechanics. The notion that the particle is both up and down at once is really just a way that people try to explain it to non-physicists. There is an exact mathematics behind everything that is much more precise.
You can imagine that a particle with spin has a corresponding 2-component vector, each component being a complex number. One complex number for each measurement outcome. For example
$$|\psi \rangle = \begin{bmatrix} 1/\sqrt{2} \\ i/\sqrt{2}\end{bmatrix}$$
The probability of getting spin up in a measurement is the magnitude of that complex number, squared. In other words, if the component is $z$, then it is $z z^*$. In this case, despite the components being different, the probability is 1/2 for either outcome (that didn't have to be the case). The vector must be normalized so that the total probability is 1.
So, there is a component corresponding to spin up, and a component corresponding to spin down, and neither one is zero. Should you interpret that as the particle being in "both spin up, and spin down, at once"? IMO no, it's not that, although there isn't really an explanation of such a state of a particle that can be explained by our everyday experience.
answered Apr 13, 2023 at 3:09