{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Angular Momentum", "refinement": "This problem involves the measurement of angular momentum components of two electrons in a coupled state, specifically focusing on the p orbital angular momentum states.", "specific_scope": "The task is to determine the joint probability of measuring specific eigenvalues for the z-components of the angular momentum of two electrons in a coupled state.", "goal": "Calculate the probability of obtaining the eigenvalues of both L_1z and L_2z as -\u0127."}, "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. Given that both electrons are in p orbitals (l=1), the possible eigenvalues for L_1z and L_2z are -\u0127, 0, and +\u0127. However, the coupled state |1, 1, 2, -1> indicates that the total angular momentum state does not allow for both individual angular momentum components to simultaneously yield -\u0127. Therefore, the joint probability of measuring both L_1z and L_2z as -\u0127 is 0."}