How to determine the probability of measuring a quantum state in state 1 using the expectation value of an operator A?
Only if the state space is two dimensional - like the state space of spin-1/2 -
you can derive the probabilities of the individual states from the expectation value of an observable A.
If the state space has higher dimension, you need more information, not only the expectation value.
If the state space is two dimensional and the system is in a general state, say |psi>, then expanding the state in the orthonormal basis of the eigenstates of the observable A, say {| psi 1>,| psi 2>}, we have |psi>=c1| psi 1>+ c2| psi 2>, where c1, c2 are the probability amplitudes of finding the system in the states | psi 1>,| psi 2>, respectively.
The expectation value of A in the state |psi> is < psi|A|psi>, and using the orthonormality of the basis {| psi 1>,| psi 2>}, the previous expectation value reads
< psi|A|psi>=| c1|2λ1+| c2|2λ2 (1),
where λ1, λ2 are the eigenvalues – i.e. the possible values – of the observable A.
Also, since the state |psi> is normalized, i.e. =1, then
| c1|2+| c2|2=1 (2)
Solving (1) and (2) for | c1|2,| c2|2, we obtain, after some elementary algebra and assuming that λ1 is different from λ2, i.e. non-degenerate eigenvalues,
| c1|2=(< psi|A|psi>- λ2)/( λ1- λ2) and | c2|2=( λ1-< psi|A|psi>)/( λ1- λ2),
which are the two probabilities.
In general you cannot.
If you average 100 numbers, and you only know the average,
how could you possibly reconstruct the 100 numbers?
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