{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Many-Particle Systems \u2192 Identical Particles in Potential Wells", "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, 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) in a one-dimensional box of length L, using the energy formula E = \u03c0\u00b2\u0127\u00b2/(2mL\u00b2).", "goal": "Calculate the specific energy levels for the ground state, first excited state, and second excited state of the system."}, "summary": "This problem requires understanding the quantum mechanics of identical fermions in a one-dimensional infinite potential well. The energies of the system are determined by the allowed quantum states, which are influenced by the Pauli exclusion principle. For four spin-1/2 particles, the ground state and excited states can be calculated based on the available quantum numbers. The goal is to find the energies of the ground state, first excited state, and second excited state in terms of the given energy E. The correct answer is found by evaluating the combinations of quantum states that the particles can occupy."}