{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Quantum Operators \u2192 Hamiltonians", "refinement": "This problem involves finding the eigenvalues of a Hamiltonian operator that is expressed in terms of Pauli matrices and a unit vector.", "specific_scope": "The Hamiltonian operator H = \u03b5\u03c3\u20d7.n\u20d7 is a linear combination of the Pauli matrices, which represent spin-1/2 systems, and the problem requires determining its eigenvalues.", "goal": "Identify the eigenvalues of the Hamiltonian operator based on the properties of the Pauli matrices and the given parameters."}, "summary": "In this problem, we are tasked with finding the eigenvalues of a Hamiltonian operator defined as H = \u03b5\u03c3\u20d7.n\u20d7, where \u03c3\u20d7 represents the Pauli spin matrices and n\u20d7 is a unit vector. The eigenvalues of this Hamiltonian can be derived from the properties of the Pauli matrices, which yield eigenvalues of \u00b11 when applied to a spin-1/2 system. Therefore, the eigenvalues of the Hamiltonian are given by \u00b1\u03b5, making the correct answer option 1: +\u03b5, -\u03b5."}