{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Measurement Theory \u2192 Probability of Measurement Outcomes", "refinement": "This problem involves understanding the relationship between quantum states, operators, and measurement outcomes in a quantum system.", "specific_scope": "The focus is on calculating the probability of measuring a specific eigenvalue of operator B after obtaining an eigenvalue from operator A, given the state vector and the operators' matrices.", "goal": "Determine the probability of measuring the eigenvalue 1 for operator B after measuring the eigenvalue 1 for operator A."}, "summary": "In this quantum mechanics problem, we analyze the measurement outcomes of two operators, A and B, on a given state vector. After measuring operator A and obtaining the eigenvalue 1, we need to find the probability of measuring the eigenvalue 1 for operator B. This involves projecting the state onto the eigenspace of operator A and then calculating the probability based on the resulting state and the eigenstates of operator B. The final answer is determined to be 1/4."}