{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Operators \u2192 Hamiltonians", "refinement": "This problem involves the analysis of a Hamiltonian operator that incorporates spin matrices and a constant related to energy.", "specific_scope": "The focus is on determining the eigenvalues of a Hamiltonian operator defined in terms of the Pauli spin matrices and a constant, which represents a physical system's energy levels.", "goal": "Identify the eigenvalues of the given Hamiltonian operator."}, "summary": "In this problem, we analyze a Hamiltonian operator defined using the Pauli spin matrices and a constant related to energy. The task is to find the eigenvalues of this operator, which represent the possible energy levels of the system. The correct eigenvalues, derived from the properties of the Pauli matrices and the structure of the Hamiltonian, are + \u03b2/2 and - \u03b2/2, indicating the quantized energy states of the system."}