{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Mechanics \u2192 Eigenvalues and Eigenvectors", "refinement": "This problem involves calculating the eigenvector of a quantum mechanical operator for a specific eigenvalue, focusing on the properties of spin operators in quantum mechanics.", "specific_scope": "The context is the calculation of the eigenvector corresponding to the eigenvalue -\u0127/2 for a muon, using the given components of the operator Q in the y-z plane.", "goal": "Determine the normalized eigenvector associated with the specified eigenvalue of the operator Q."}, "summary": "This problem is rooted in quantum mechanics, specifically in finding the eigenvector of a quantum operator corresponding to a given eigenvalue. The operator Q has components defined in the y-z plane, and the task is to calculate the eigenvector for the eigenvalue -\u0127/2. The solution involves understanding the properties of the matrices representing the operator components and applying the appropriate mathematical techniques to find the normalized eigenvector. The correct answer is (sin(\u03c6/2), -cos(\u03c6/2)), which represents the state of the muon along the specified direction."}