{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Spin Systems", "refinement": "This problem involves calculating the expectation value of a spin operator for a quantum state represented in a specific basis.", "specific_scope": "The focus is on finding the expectation value of the spin operator along the y-direction (S_y) for an electron in a given spin state.", "goal": "Calculate the expectation value of S_y using the provided spin state and the corresponding spin operator matrix."}, "summary": "This problem is rooted in quantum mechanics, specifically in the study of spin systems. It requires calculating the expectation value of the spin operator along the y-direction for an electron in a given spin state. The spin state is represented as a vector, and the spin operator S_y is given as a matrix. The goal is to compute the expectation value by applying the operator to the state vector and using the principles of quantum mechanics. The correct answer is -12*hbar/25."}