{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Linear Algebra \u2192 Eigenvalues and Eigenvectors", "refinement": "This problem involves finding the eigenvector of a quantum mechanical operator corresponding to a specific eigenvalue, using the properties of matrices and their eigenvectors.", "specific_scope": "The focus is on the operator P for a muon, with its components defined in terms of 2x2 matrices, and the task is to find the normalized eigenvector for a specific direction in the x-z plane.", "goal": "Calculate the normalized eigenvector corresponding to the eigenvalue +\u0127/2 for the operator P along an arbitrary direction in the x-z plane."}, "summary": "This problem requires calculating the eigenvector of a quantum mechanical operator for a muon, specifically along an arbitrary direction in the x-z plane. The operator P is defined by its components P_x, P_y, and P_z, represented by 2x2 matrices. The goal is to find the normalized eigenvector corresponding to the eigenvalue +\u0127/2. The correct answer is the normalized eigenvector (cos(\u03b8/2), sin(\u03b8/2)), which reflects the quantum state of the system in the specified direction."}