{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Spin Systems", "refinement": "This problem involves calculating the expected value of a quantum mechanical spin component, specifically in the y-direction, using the spin state vector of an electron.", "specific_scope": "The focus is on the spin state vector given in the form of complex coefficients, which represent the quantum state of the electron, and the application of the appropriate operator to find the expected value of the spin component.", "goal": "Calculate the expected value of the spin component S_y for the given spin state vector."}, "summary": "In this problem, we are tasked with finding the expected value of the spin component in the y-direction (S_y) for an electron described by a specific spin state vector. This involves using quantum mechanics principles to apply the S_y operator to the state vector and compute the expected value. The result of the calculation yields -12\u0127/25, which corresponds to the expected value of the spin component in the y-direction."}