Show that, assuming |9₁) and ₂) are properly normalized, [4₁) and [₂) are orthonormal. That means show that they are orthogonal to each other and that each one of them is normalized. a) 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 the system immediately after this measurement? c) Immediately after the measurement of A (which, recall, happened to yield a₁), the observable B is then measured. What are the possible results of the B measurement, and what are their probabilities?
Show that, assuming |9₁) and ₂) are properly normalized, [4₁) and [₂) are orthonormal. That means show that they are orthogonal to each other and that each one of them is normalized. a) 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 the system immediately after this measurement? c) Immediately after the measurement of A (which, recall, happened to yield a₁), the observable B is then measured. What are the possible results of the B measurement, and what are their probabilities?
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I need some help with this quantum mechanics question.
Transcribed Image Text:1. More practice with quantum measurements
An operator A (representing observable A) has two normalized eigenstates ₁) and ₂), with
eigenvalues ai and a2, respectively. Operator B (representing observable B) has two normalized
eigenstates ₁) and 2), with eigenvalues b₁ and b₂. Suppose these eigenstates are related by
the following:
√8
1
√8
13|9₂), 14 ) 2 = 3|41) - 12)
14/₁) = 3/8₁₂) +-
a)
Show that, assuming |9₁) and 2) are properly normalized, ₁) and 2) are
orthonormal. That means show that they are orthogonal to each other and that each one of them
is 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 the
system immediately after this measurement?
c)
Immediately after the measurement of A (which, recall, happened to yield a₁), the
observable B is then measured. What are the possible results of the B measurement, and what are
their 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, we
measure A once again. What is the probability of getting outcome al again? Hint: If you think the
answer 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 has
been measured, nature knows the result, but YOU do not. So as far as you are concerned, there
are two possibilities, b₁ or b2, with relative probabilities given in part c above.) If you now
remeasure 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 measured
A again, right away. What are the possible result(s) of the second measurement of A, with what
probabilities?
- iv) Do you think the operators A and B commute? Why/why not?
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