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I'm a little unclear on what the Many Worlds Interpretation (or family of interpretations), MWI, says will happen instead of a collapse.

Suppose I measure particle spin on the X-axis. The Copenhagen Interpretation, CI, would say that the particle was in a superposition of spin up and spin down for the X-axis before the measurement was made; that the measurement process collapsed that superposition; and, that one of the values possible for that sort of experiment will randomly occur.

Does MWI say that we will get (A) two branches of the universe (one in which the value of spin-up occurred and one in which the value of spin-down occurred as a result of the measurement) or (B) a much larger number of branches (one for each possible result of each possible experiment that could have been done on the given quantum system)?

If the answer is (A), what prevents (B) from happening?

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The answer is (A) (unless there's another branch of the universal wave function in which you perform a different experiment).

I'm not sure that you mean by "what prevents (B) from happening?". The Schrodinger equation? The MWI doesn't say that "Anything that could logically happen, does." You still have some initial wave function that the Schrodinger equation deterministically takes to another wave function, and the final wave function won't have any components whose "ingredients" weren't in the initial one.

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