{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Operators and Uncertainty", "refinement": "This problem involves calculating the uncertainty of an observable (P_z) when the system is in a specific quantum state (eigenstate of P_x).", "specific_scope": "The focus is on determining the uncertainty \u0394P_z using the properties of the operators and the given state vector in the context of quantum mechanics.", "goal": "Calculate the uncertainty \u0394P_z of the operator P_z when the system is in the specified eigenstate of P_x."}, "summary": "In this quantum mechanics problem, we need to calculate the uncertainty \u0394P_z of the operator P_z when the system is in a specific eigenstate of P_x. The state vector is given, and we can use the properties of the operators and the state to find the expectation values needed to compute the uncertainty. The uncertainty is determined by the formula \u0394P_z = \u221a(\u27e8P_z^2\u27e9 - \u27e8P_z\u27e9^2). After performing the necessary calculations, we find that the uncertainty \u0394P_z is \u0127/\u221a(2), which corresponds to option 1."}