{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Spin Systems \u2192 Measurement and Expectation Values", "refinement": "This problem involves calculating probabilities of measuring a quantum state in specific eigenstates and finding the expectation value of a given operator.", "specific_scope": "The focus is on an electron's spin state represented in terms of its eigenstates and the associated operator defined by its matrix elements.", "goal": "Calculate the probabilities of measuring the electron in the |up> and |down> states, and find the average value of the specified operator."}, "summary": "In this quantum mechanics problem, we analyze the state of an electron described by a superposition of spin states |up> and |down>. We need to calculate the probabilities of measuring the electron in these states and find the expectation value of a specific operator defined by its matrix elements. The solution involves normalizing the state vector, computing the probabilities, and using the operator's matrix representation to find the average value. The correct answer is the probabilities 0.64 and 0.36, along with the expectation value hbar / 7."}