{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Many-Particle Systems \u2192 Quantum States in Potential Wells", "refinement": "This problem involves calculating the energy levels of a system of identical spin-1/2 particles in a one-dimensional infinite potential well, taking into account the principles of quantum mechanics and the Pauli exclusion principle.", "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) confined in a one-dimensional box of length L.", "goal": "Calculate the specific energy levels for the ground state, first excited state, and second excited state of the system in terms of the given energy E."}, "summary": "This problem requires an understanding of quantum mechanics, specifically the behavior of identical fermions in a one-dimensional infinite potential well. The task is to calculate the energy levels for the ground state and excited states of a system of four spin-1/2 particles, considering the Pauli exclusion principle. The energies are expressed in terms of a base energy E, leading to specific values for the ground state and excited states. The correct energy levels must be determined from the provided options."}