{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Operators and Uncertainty", "refinement": "This problem involves calculating the uncertainty of an observable (Q_z) in a quantum state that is an eigenstate of another observable (Q_x).", "specific_scope": "The focus is on determining the uncertainty \u0394Q_z for the operator Q_z when the system is in a specific eigenstate of Q_x.", "goal": "Calculate the uncertainty \u0394Q_z using the properties of the quantum state and the operator Q_z."}, "summary": "In this problem, we are tasked with finding the uncertainty \u0394Q_z of the operator Q_z when the system is in an eigenstate of Q_x. The eigenstate is represented by the column vector (-1/\u221a3, 1/\u221a3, -1/\u221a3). To find the uncertainty, we need to calculate the expectation value of Q_z and the expectation value of Q_z squared, then use these to find the uncertainty. The result shows that the uncertainty \u0394Q_z is \u0127/\u221a3."}