{"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 of angular momentum operators for two electrons in a coupled state.", "specific_scope": "The task is to determine the probability of obtaining specific eigenvalues for the z-components of the angular momentum operators for two electrons in a p orbital state, given their total angular momentum state.", "goal": "Calculate the joint probability of measuring the eigenvalues of both L_1z and L_2z as -\u0127."}, "summary": "In this quantum mechanics problem, we are examining 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 means their individual orbital angular momentum quantum numbers are both 1. The total angular momentum state is given as |1, 1, 2, -1>. To find the joint probability of measuring both L_1z and L_2z as -\u0127, we need to analyze the possible outcomes based on the quantum state and the allowed values of the z-components. The correct answer is determined by the allowed combinations of the individual angular momentum states."}