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Suppose I have states $|1\rangle$ and $|2\rangle$, and my system is in a quantum mixed state \begin{equation} c \left( |1\rangle + \sqrt{3} |2\rangle \right). \end{equation} In a first measurement I get the first state $|1\rangle$. So how is it impossible to get the second state in the next successive measurement? I mean the total probability is $1$ and because the probability of getting individual states is not $0$ then how can the probability of getting two different states in successive measurements of this system be $0$?

Moreover, can the probability of a getting a particular state which is the combination of two successive measurement of states ever be $0$?

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It's not entirely clear what you're asking, but it seems like maybe you're confusing two separate things. Before you make your first measurement, you assume that you're in this superposition of states. What you're apparently telling us is that there are precisely two possible outcomes that could occur: either $|1 \rangle$, or $|2\rangle$. Not some weird combination of these two or some other $|3 \rangle$ — but just one of those two possibilities. You can also calculate the probability of getting either of those outcomes: a 25% chance of $|1 \rangle$ and a 75% chance of $|2\rangle$.

But that's all before you make a measurement. Now, measurement does something weird and discontinuous and surprising. It changes the wavefunction suddenly, in a way that doesn't conserve individual probabilities. The fact that you measured the system and found it to be in state $|1\rangle$ means that all of a sudden you know that the probabilities are different: it's 100% in $|1\rangle$ and 0% in $|2\rangle$. So if you immediately follow that observation with another observation, there's no chance that you'll get $|2\rangle$. This is called "wavefunction collapse", and — at least at an elementary level — it's generally thought of as being exactly the same thing as a measurement.

Of course, it's certainly possible that this system that was definitely only in state $|1\rangle$ once you made your measurement changes after you've made your measurement — it can evolve to be in other states, possibly back to that original mixed state you mentioned. You'll see this when you study Schrödinger's equation, which is a smoother process. So, for example, if you wait a few seconds after your first measurement to make another measurement, you may find that the second measurement does indeed produce $|2\rangle$. But in this idealized and simplified case that you're talking about, you simply define the measurement to be something that says your state was a pure $|1\rangle$ at the very instant of your observation.

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