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An operator Â, representing observable A, has two normalized eigenstates 1) and 2), with eigenvalues a₁ and a2, respectively. Another operator B, representing observable B, has two normalized eigenstates |01) and |02), with eigenvalues b₁ and b₂, respectively. The eigenstates are related by (a) 3 4 |21) = 3 1102), 142)=201010-1102) Suppose that the observable A of the system is measured and the eigenvalue a₁ is obtained. What is the state of the system immediately after this measurement? (b) Suppose that after the first measurement, the observable B is now measured. What are the possible results, and what are their probabilities? (c) Right after measurement of B, the observable A is measured again. What is the probability of getting a₁? What would be the answer if instead the observable B is not measured? (d) Construct the spectral decomposition of operators  and B. Show that two operators do not commute. Note: The measurement described above is called a sequential measurement. For two incompatible observables that are measured sequentially, the two measurements disturb one another.

An operator Â, representing observable A, has two normalized eigenstates 1) and 2), with eigenvalues a₁ and a2, respectively. Another operator B, representing observable B, has two normalized eigenstates |01) and |02), with eigenvalues b₁ and b₂, respectively. The eigenstates are related by (a) 3 4 |21) = 3 1102), 142)=201010-1102) Suppose that the observable A of the system is measured and the eigenvalue a₁ is obtained. What is the state of the system immediately after this measurement? (b) Suppose that after the first measurement, the observable B is now measured. What are the possible results, and what are their probabilities? (c) Right after measurement of B, the observable A is measured again. What is the probability of getting a₁? What would be the answer if instead the observable B is not measured? (d) Construct the spectral decomposition of operators  and B. Show that two operators do not commute. Note: The measurement described above is called a sequential measurement. For two incompatible observables that are measured sequentially, the two measurements disturb one another.

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An operator Â, representing observable A, has two normalized eigenstates 1) and 2), with eigenvalues a₁ and a2, respectively. Another operator B, representing observable B, has two normalized eigenstates |01) and |02), with eigenvalues b₁ and b₂, respectively. The eigenstates are related by (a) 3 4 |21) = 3 1102), 142)=201010-1102) Suppose that the observable A of the system is measured and the eigenvalue a₁ is obtained. What is the state of the system immediately after this measurement? (b) Suppose that after the first measurement, the observable B is now measured. What are the possible results, and what are their probabilities? (c) Right after measurement of B, the observable A is measured again. What is the probability of getting a₁? What would be the answer if instead the observable B is not measured? (d) Construct the spectral decomposition of operators  and B. Show that two operators do not commute. Note: The measurement described above is called a sequential measurement. For two incompatible observables that are measured sequentially, the two measurements disturb one another.
Transcribed Image Text:An operator Â, representing observable A, has two normalized eigenstates 1) and 2), with eigenvalues a₁ and a2, respectively. Another operator B, representing observable B, has two normalized eigenstates |01) and |02), with eigenvalues b₁ and b₂, respectively. The eigenstates are related by (a) 3 4 |21) = 3 1102), 142)=201010-1102) Suppose that the observable A of the system is measured and the eigenvalue a₁ is obtained. What is the state of the system immediately after this measurement? (b) Suppose that after the first measurement, the observable B is now measured. What are the possible results, and what are their probabilities? (c) Right after measurement of B, the observable A is measured again. What is the probability of getting a₁? What would be the answer if instead the observable B is not measured? (d) Construct the spectral decomposition of operators  and B. Show that two operators do not commute. Note: The measurement described above is called a sequential measurement. For two incompatible observables that are measured sequentially, the two measurements disturb one another.
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