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I've read that once a measurement prepares a quantum state, subsequent measurements confirm it. After all, projecting a basis vector onto itself is an identity.

But if an electron's spin is first prepared in a horizontal magnetic field, and then measured in a vertical magnetic field, then it will either: emit a photon (down state) or not emit a photon (up state), with equal probability. Subsequent measurements in the vertical magnetic field should confirm the first measurement, but if a photon were emitted in the first measurement, then it would not be emitted again because the first emission would cause the electron to switch to the up state. So the first measurement will show down, but every subsequent measurement will show up.

Somewhere I'm confused.

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It sounds like your thinking that there is a fixed number of photons. Only 1 photon and so no more could be emitted. Like some fixed number of classical balls. Remember the light wave you're using to measure is all photons. Suppose you wanted a qualitative answer unlike the quantitative answer of the comment.

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  • $\begingroup$ The spin up state is the lower energy state. The photon that gets emitted is equal to the difference in energy between the down and up state. So when the first measurement results in a photon, the spin must have transitioned from down to up. In subsequent measurements, it has no more energy to lose, so no more photons appear. $\endgroup$
    – Tomek Dobrzynski
    Commented Jan 29, 2018 at 1:19

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