{"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 states.", "specific_scope": "The task is to calculate the joint probability of obtaining specific eigenvalues for the angular momentum operators L_1z and L_2z for a two-electron system in a given coupled state.", "goal": "Determine the probability of measuring both L_1z and L_2z as -\u0127."}, "summary": "In this quantum mechanics problem, we are analyzing the joint probability of measuring the z-components of angular momentum for two electrons in a coupled state. The electrons are in p orbitals, which have an orbital angular momentum quantum number of 1. The coupled state |1, 1, 2, -1> indicates that the total angular momentum is 2 with a z-component of -1. To find the joint probability of measuring both L_1z and L_2z as -\u0127, we need to consider the possible values of L_1z and L_2z that can yield the total z-component of -1. The calculations lead us to determine that the joint probability is 1/2."}