{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Quantum States \u2192 Expectation Values", "refinement": "This problem involves calculating the expectation value of a linear combination of quantum operators for a spin-1/2 particle in a given superposition state.", "specific_scope": "The focus is on the expectation value of the operator 10\u03c3z + 5\u03c3x for a specific quantum state represented as a linear combination of the eigenstates |\u2191\u27e9 and |\u2193\u27e9.", "goal": "Calculate the expectation value of the operator for the given quantum state and round the result to one decimal place."}, "summary": "In this problem, we are tasked with calculating the expectation value of the operator 10\u03c3z + 5\u03c3x for a spin-1/2 particle in a superposition state. The state is given as a combination of the eigenstates |\u2191\u27e9 and |\u2193\u27e9. To find the expectation value, we need to apply the operators \u03c3z and \u03c3x to the state, compute the resulting values, and then combine them according to the coefficients in the operator. The final result, rounded to one decimal place, is -0.7."}