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I want to ask how do we actually measure the probability amplitude that appears in Schrödinger equation. From what I read in quantum mechanics textbooks, it appears that after the measurement, the system "collapses" to its eigen-state. And we know this because the probability amplitude of events changed. For a typical event involving two states, previously we only know its amplitude is $|A_{1}+A_{2}|^{2}$, but now it is $|A_{1}|^{2}+|A_{2}|^{2}$, etc.

My question is, how do we know that our measurement of the probability amplitude is accurate? How do we know that there is no uncertainity principle that makes $$ \Delta A_{1}*\Delta A_{2}\ge k $$ for example? Since the probability amplitude is one of the intrinsic quantum properties of the system, it seems to me any measurement should disturb it to some extent. For example, if the above hypothetical "uncertainty relation" holds, then we cannot say in principle what $|A_1+A_2|^{2}$ is (since it must be disturbed by our measurement), but only what $E|A_1+A_2|^{2}$ is based on experiment. But if $A_1,A_{2}$ is in certain range, we may not be able to distinguish $|A_1+A_2|$ and $|A_1|^{2}+|A_2|^{2}$ anymore unless we take a huge number of experiments.

To elaborate it, my rough conception of the way people measure it is this: We make the experiment in identical situation $N$ times, and we assume via strong law of large numbers that the average frequency must approach the mean value. However, this assumption does not exclude the possibility that every time we measure the probability of event $A_1$, the accuracy of measuring event $A_2$ may be somehow influenced. Therefore in actual measurement, the probability we get is close to $E|A_1+A_2|^{2}$, but not really the real value if by measurement we caused a huge variance in the data.

Let us for simplicity consider an even simpler case with only one event $A$. $A$'s probability amplitude is given by the complex number $a$. Suppose in actual measurement, we found if we take $N$ measurements at the $i$th time, then the sample average is about $a+(-1)^{i}b$. Then we can propose either $A$'s probability amplitude changes with time (like in a two state system), or our measurement somehow influenced $a$'s sample value. Suppose we are in the second case, how do we know what is the true probability amplitude $a$? If we only do $N$ experiments, we would only get a biased value. And if we do more, the chance of the bias accumulating is small but still not negligible if $b$ is really large.

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You're asking a couple of different questions here. In the last paragraph you're just asking how you can tell that the measurements done on an ensemble of presumably identically prepared systems are uncorrelated. In practice you would vary the time interval at which you do the measurements to check for time correlation. –  DanielSank May 26 '14 at 5:04

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