{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Quantum Operators \u2192 Hamiltonians", "refinement": "This problem involves the analysis of a Hamiltonian operator that incorporates spin and an external direction defined by a unit vector.", "specific_scope": "The focus is on determining the eigenvalues of the Hamiltonian operator defined as H = \u03b5\u03c3\u20d7.n\u20d7, where \u03c3\u20d7 represents the Pauli matrices and n\u20d7 is a unit vector.", "goal": "Identify the eigenvalues of the Hamiltonian operator based on its mathematical structure and the properties of the Pauli matrices."}, "summary": "In this problem, we analyze a Hamiltonian operator defined as H = \u03b5\u03c3\u20d7.n\u20d7, where \u03b5 is a constant energy parameter and \u03c3\u20d7 represents the Pauli spin matrices. The eigenvalues of this Hamiltonian can be determined by recognizing that the Pauli matrices have eigenvalues of +1 and -1. Therefore, when multiplied by the constant \u03b5, the eigenvalues of the Hamiltonian are +\u03b5 and -\u03b5. Thus, the correct answer is +\u03b5, -\u03b5."}