{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Many-Body Systems \u2192 Spin Systems", "refinement": "This problem involves calculating the ground state energy of a system of identical spin-1/2 particles in a harmonic oscillator potential, taking into account the effects of quantum statistics (Fermi-Dirac statistics for fermions).", "specific_scope": "The focus is on a system of three identical fermions (spin-1/2 particles) in a three-dimensional isotropic harmonic oscillator potential, where the Pauli exclusion principle applies.", "goal": "Determine the total ground state energy of the system in terms of the given energy scale E_0."}, "summary": "This problem is rooted in quantum mechanics, specifically in the study of many-body systems of identical fermions. It requires calculating the ground state energy of three spin-1/2 particles in a three-dimensional isotropic harmonic oscillator potential, considering the Pauli exclusion principle. The total energy of the system is expressed in terms of a reference energy scale, E_0. The correct answer, 4.5E_0, reflects the unique energy levels available to the fermions due to their indistinguishable nature and the constraints imposed by quantum statistics."}