{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Many-Particle Systems \u2192 Quantum States of Identical Particles", "refinement": "This problem involves calculating the energy levels of a system of identical spin-1/2 particles confined in a one-dimensional infinite potential well.", "specific_scope": "The focus is on determining the ground state and excited state energies for a system of four identical fermions (spin-1/2 particles) in a one-dimensional box, taking into account the Pauli exclusion principle.", "goal": "Calculate the energies of the ground state, first excited state, and second excited state of the system in terms of a given energy E."}, "summary": "This problem is rooted in quantum mechanics, specifically in the behavior of identical fermions in a confined space. The task is to calculate the energy levels for a system of four spin-1/2 particles in a one-dimensional infinite potential well. The energies are derived from the allowed quantum states, considering the Pauli exclusion principle, which dictates that no two fermions can occupy the same quantum state. The resulting energies for the ground state, first excited state, and second excited state are found to be 10E, 15E, and 18E, respectively."}