{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Spin Systems \u2192 Hamiltonians", "refinement": "This problem involves calculating the eigenvalues of a Hamiltonian operator for a quantum system, specifically for a spin-1/2 particle in a magnetic field.", "specific_scope": "The Hamiltonian is defined in terms of the magnetic field and the spin operator for the y-component, requiring the use of matrix algebra to find the eigenvalues.", "goal": "Determine the eigenvalues of the Hamiltonian operator H to understand the energy levels of the spin-1/2 particle in the magnetic field."}, "summary": "This problem focuses on quantum mechanics, specifically the calculation of eigenvalues for a Hamiltonian operator representing a spin-1/2 particle in a magnetic field. The Hamiltonian is given as H = -\u03b3B_y S_y, where S_y is the spin operator for the y-component. By calculating the eigenvalues of the matrix representation of the Hamiltonian, we find that the eigenvalues are \u00b1\u03b3\u0127B_y/2, indicating the energy levels of the system in the presence of the magnetic field."}