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I'm confused about the state of a system after a measurement. Say we have a particle $v$ in the state: $ |\psi\rangle= \sqrt{1/4} \ |0\rangle + \sqrt{3/4} \ |1\rangle $.

From my understanding, if one were to measure the state of $v$, one would get the result $|0\rangle$ with probability $|\sqrt{1/4}|^2=1/4$, and similarly, $|1\rangle$ with probability $3/4$.

However, I've also learned that a measurement is always done by an observable (a unitary operator), e.g. $Z=|0\rangle \langle 0|-|1\rangle \langle 1|$, and that the outcome of the measurement is an eigenvalue of this operator, and that the state we get after the measurement is always dependent on the observable we use, and similarly for the probability of getting that state.

Now, by inspection, I noticed that when I measure $Z$, I do get the state $|0\rangle$ with probability $1/4$, and $|1\rangle$ with probability $3/4$, as expected. But I don't get these results when I measure the Pauli operator $X$, for example.

Does that mean that the claim in my second paragraph always assumes a measurement of $Z$?

4Observables are Hermitian operators, not (necessarily) unitary ones. – tparker – 2020-05-22T12:16:37.213