Figure 8.3 shows the Stern-Gerlach apparatus. It reveals that spin-½ particles have just two possible spin states. Assume that when these two beams are separated inside the channel (though still near its centerline), we can choose to block one or the other for study. Now a second such apparatus is added after the first. Their channels are aligned, but the second one is rotated about the x'-axis by an angle ➢ from the first. Suppose we block the spin-down beam in the first apparatus, allowing only the spin-up beam into the second. There is no wave function for spin, but we can still talk of a probability amplitude, which we square to give a probability. After the first apparatus' spin-up beam passes through the second apparatus, the probability amplitude is cos (➢ /2) ↑ T2nd + sin (➢ /2) ↓2nd, where the arrows indicate the two possible findings for spin in the second apparatus, (a) What is the probability of finding the particle spin up in the second apparatus? Of finding it spin down? Argue 4hat these probabilities make sense individually for representative values of ø and that their sum is also sensible, (b) By contrasting this spin probability amplitude with a spatial probability amplitude, such as ψ(x)= Ae-bx2, argue that although the arbitrariness of ➢ gives the spin case an infinite number of values, it is still justified to refer to it as a "two-state system," while the spatial case is an infinite-state system.
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