{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Atomic Physics \u2192 Angular Momentum", "refinement": "This problem pertains to the quantization of angular momentum in quantum systems, specifically focusing on the orbital angular momentum of a hydrogen atom.", "specific_scope": "The context involves calculating the probability of measuring a specific eigenvalue of the orbital angular momentum component L_z for a hydrogen atom in a given quantum state defined by quantum numbers n, l, and m.", "goal": "Determine the probability of obtaining the eigenvalue 0 \u0127 for the L_z measurement based on the quantum state of the hydrogen atom."}, "summary": "In this problem, we analyze the orbital angular momentum of a hydrogen atom characterized by quantum numbers n = 3, l = 2, and m = 1. The measurement of the L_z component yields quantized eigenvalues determined by the magnetic quantum number m. Since m = 1, the possible eigenvalues for L_z are given by m\u0127, which in this case is 1\u0127. Therefore, the probability of measuring L_z = 0\u0127 is 0, as the state does not allow for that eigenvalue."}