{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Spin Systems \u2192 Expectation Values", "refinement": "This problem involves calculating the expectation value of a linear combination of spin operators for a spin-half particle in a given quantum state.", "specific_scope": "The state of the spin-half particle is expressed as a superposition of the eigenstates of the \u03c3z operator, and the expectation value is calculated for the operator 10\u03c3z + 5\u03c3_x.", "goal": "Determine the expectation value of the operator for the given quantum state."}, "summary": "In this problem, we need to calculate the expectation value of the operator 10\u03c3z + 5\u03c3_x for a spin-half particle in a specific superposition state. The state is given as a combination of the spin-up and spin-down states, which are eigenstates of the \u03c3z operator. To find the expectation value, we will apply the operators to the state and compute the resulting values. The goal is to arrive at a numerical result rounded to one decimal place."}