Two-level quantum system observable quantities

[crowlma]

Homework Statement

A two-level system is spanned by the orthonormal basis states $|a_{1}>$ and $|a_{2}>$. The operators representing two particular observable quantities A and B are:
$\hat{A} = α(|a_{1}> <a_{1}| - |a_{2}> <a_{2}|)$
and $\hat{B} = β(|a_{1}> <a_{2}| + |a_{2}> <a_{1}|)$

a) The state of the system is $|\Psi> = |a_{1}>$. If you measured A, what result/s would you get and what is the probability of obtaining each of these results? If you measured B, what results would you get and what is the probability of obtaining each of these results? What is the expectation value and the uncertainty of A and B?

b) The state of the system is $|\Psi> = \frac{1}{\sqrt{2}}(|a_{1}> + |a_{2}>)$. If you measured A, what result/s would you get and what is the probability of obtaining each of these results? If you measured B, what results would you get and what is the probability of obtaining each of these results? What is the expectation value and the uncertainty of A and B?

c) Are A and B compatible observables? Explain your reasoning?

Homework Equations

for part c): The commutator $[\hat{A},\hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A}$

The Attempt at a Solution

Ok so we did this in class and I can follow the working somewhat in parts, but looking for a more complete grasp.

Since A is an observable, measuring A in the system state $|a_{1}>$ gives $\hat{A}|a_{1}>$ which comes out to $α(|a_{1}> <a_{1}| - |a_{2}> <a_{2}|)|a_{1}> = α(|a_{1}> <a_{1}|a_{1}> - |a_{2}> <a_{2}|a_{1}>) = α|a_{1}>$. So α is a result for A. This I understand. Then I'm not sure why, in class, we did the same thing but substituted in |a₂> as the system state, giving -α|a₂> as the result. This makes -α another result for A. So these are the two results that are gettable, then $p(α)= |<a_{1}|ψ>|^{2} = |<a_{1}|a_{1}>|^{2} = 1$. Makes sense, then subsequently p(-α) = 1- p(α) = 0.

Then finding the results and probability for B, I don't really understand. I tried to do it the same as for A and got β as the only possible result, but the result is both positive and negative β.

I am fine finding the probabilities, just struggle with finding the results of the observables at the moment.

For part c) it is my understanding that if A and B commute then they are compatible. $[\hat{A},\hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A} = 0$ if they commute. I keep getting that they commute however in class we found that they do not/that they are not compatible, so I'm a bit confused here.Any help on any parts of the above would be greatly appreciated!

 

[crowlma]

Sorry to bump but is anyone able to provide any assistance at all?