I need some help with this quantum mechanics question. 1. More practice with quantum measurementsAn operator A (representing observable A) has two normalized eigenstates ₁) and ₂), witheigenvalues ai and a2, respectively. Operator B (representing observable B) has two normalizedeigenstates ₁) and 2), with eigenvalues b₁ and b₂. Suppose these eigenstates are related bythe following:√81√813|9₂), 14 ) 2 = 3|41) - 12)14/₁) = 3/8₁₂) +-a)Show that, assuming |9₁) and 2) are properly normalized, ₁) and 2) areorthonormal. That means show that they are orthogonal to each other and that each one of themis normalized.b) Let's start in some unspecified random state, and then observable A is measured. Further,assume that the result of the measurement is the particular value a₁. What is the state of thesystem immediately after this measurement?c)Immediately after the measurement of A (which, recall, happened to yield a₁), theobservable B is then measured. What are the possible results of the B measurement, and what aretheir probabilities?d)Consider three scenarios following up on the "setup" in part C:- i) Suppose I tell you that, when we measured B, we found b₁. And immediately after that, wemeasure A once again. What is the probability of getting outcome al again? Hint: If you think theanswer is trivially "1" or trivially "1/2", I suggest you think again!- ii) Suppose instead that you measure B, but you do not know the outcome! (To be clear: B hasbeen measured, nature knows the result, but YOU do not. So as far as you are concerned, thereare two possibilities, b₁ or b2, with relative probabilities given in part c above.) If you nowremeasure A, what is the probability of getting ai?- iii) Go back a step-suppose that after our very first measurement of A (when, you will recall,we happened to find a₁), we had completely neglected to measure B at all, and simply measuredA again, right away. What are the possible result(s) of the second measurement of A, with whatprobabilities?- iv) Do you think the operators A and B commute? Why/why not?

Question

I need some help with this quantum mechanics question. 1. More practice with quantum measurementsAn operator A (representing observable A) has two normalized eigenstates ₁) and ₂), witheigenvalues ai and a2, respectively. Operator B (representing observable B) has two normalizedeigenstates ₁) and 2), with eigenvalues b₁ and b₂. Suppose these eigenstates are related bythe following:√81√813|9₂), 14 ) 2 = 3|41) - 12)14/₁) = 3/8₁₂) +-a)Show that, assuming |9₁) and 2) are properly normalized, ₁) and 2) areorthonormal. That means show that they are orthogonal to each other and that each one of themis normalized.b) Let's start in some unspecified random state, and then observable A is measured. Further,assume that the result of the measurement is the particular value a₁. What is the state of thesystem immediately after this measurement?c)Immediately after the measurement of A (which, recall, happened to yield a₁), theobservable B is then measured. What are the possible results of the B measurement, and what aretheir probabilities?d)Consider three scenarios following up on the &#34;setup&#34; in part C:- i) Suppose I tell you that, when we measured B, we found b₁. And immediately after that, wemeasure A once again. What is the probability of getting outcome al again? Hint: If you think theanswer is trivially &#34;1&#34; or trivially &#34;1/2&#34;, I suggest you think again!- ii) Suppose instead that you measure B, but you do not know the outcome! (To be clear: B hasbeen measured, nature knows the result, but YOU do not. So as far as you are concerned, thereare two possibilities, b₁ or b2, with relative probabilities given in part c above.) If you nowremeasure A, what is the probability of getting ai?- iii) Go back a step-suppose that after our very first measurement of A (when, you will recall,we happened to find a₁), we had completely neglected to measure B at all, and simply measuredA again, right away. What are the possible result(s) of the second measurement of A, with whatprobabilities?- iv) Do you think the operators A and B commute? Why/why not?

Accepted Answer

a. The state is given as|Ψ1⟩=1/3|ϕ1⟩+√8/3|ϕ2⟩|Ψ2⟩=√8/3|ϕ1⟩−1/3|ϕ2⟩the normalization…