{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Quantum States \u2192 Eigenvalues and Eigenvectors", "refinement": "This problem involves finding the eigenvector of a quantum operator corresponding to a specific eigenvalue, which is a common task in quantum mechanics when dealing with spin systems.", "specific_scope": "The focus is on determining the normalized eigenvector of the quantum operator Q for a muon in the x-z plane, specifically for the eigenvalue +\u0127/2.", "goal": "Identify the correct normalized eigenvector that corresponds to the given eigenvalue for the specified quantum operator."}, "summary": "In this problem, we are tasked with finding the normalized eigenvector of a quantum operator Q for a muon, corresponding to the eigenvalue +\u0127/2. The operator components are given in matrix form, and the solution involves recognizing that the eigenvector can be expressed in terms of angles related to the direction in the x-z plane. The correct normalized eigenvector is (cos(\u03b8/2), sin(\u03b8/2)), which reflects the quantum state of the muon along the specified direction."}