{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Angular Momentum", "refinement": "This problem involves the measurement of angular momentum in a quantum system, specifically focusing on the joint probabilities of measuring specific eigenvalues for two coupled angular momentum states.", "specific_scope": "The scenario considers two electrons in a d orbital state, represented by a coupled angular momentum state, and examines the probabilities of measuring their z-components of angular momentum.", "goal": "Determine the joint probability of measuring both L_1z and L_2z as 0\u0127 for the given coupled state."}, "summary": "In this quantum mechanics problem, we analyze the joint probability of measuring the z-components of angular momentum for two electrons in a coupled state. The state |2, 2, 3, 0> indicates that both electrons are in the maximum m state for their orbital angular momentum, which means that measuring L_1z and L_2z as 0\u0127 is not possible. Therefore, the joint probability of obtaining both measurements as 0\u0127 is 0."}