Switching observers in a quantum measurement

[Keru]

Homework Statement

(c is a constant)

The attempt at a solution

-In the first measure we got a1, so the state of the system would be psi1.
-In the second measure, there's no information about what eigenvalue we got. Would the state of the system still be psi1? Psi1 is written in terms of B eigenvectors, and as we don't know which one we measured, the state should be described as a superposition of both vectors, which is precisely psi1. Is that correct?
-In the third measure, i wrote the "betas" in terms of the "psis", so i have something like:
(A and B being constants, not the observables)
psi1 = A beta1 + B beta2 = C psi1 + D psi2 + E psi 1 + F psi2 = G psi1 + H psi 2

Is it H the coefficient that tells me the probabilities of getting the a2 eigenvalue, or did I do something wrong?

 

[kith]

-In the first measure we got a1, so the state of the system would be psi1.

Correct.

-In the second measure, there's no information about what eigenvalue we got. Would the state of the system still be psi1? Psi1 is written in terms of B eigenvectors, and as we don't know which one we measured, the state should be described as a superposition of both vectors, which is precisely psi1. Is that correct?

No. I bolded the part which is wrong. It contradicts the postulate that after a measurement, the system is in an eigenstate of the corresponding observable.

If you perform a measurement and don't have information about the outcome, you can't use a single state vector to describe the situation after the measurement. You need to do a case-by-case analysis in order to calculate the probabilities.

 

[Keru]

Ok I think i got it. So, for one of the two possible cases it would continue like this?

 

[kith]

Yes. I haven't checked the calculations but conceptually, it is correct now.