{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Mechanics \u2192 Quantum Harmonic Oscillator", "refinement": "This problem pertains to the energy levels and eigenfunctions of a quantum harmonic oscillator in three dimensions, characterized by a quadratic potential.", "specific_scope": "The focus is on determining the energy of the third excited state and the number of independent eigenfunctions corresponding to that energy level in a three-dimensional isotropic harmonic oscillator.", "goal": "Calculate the energy of the third excited state and identify the number of independent eigenfunctions associated with that energy."}, "summary": "This problem involves quantum mechanics, specifically the quantum harmonic oscillator in three dimensions. The energy levels of such a system are quantized and can be expressed in terms of the angular frequency \u03c9. The third excited state corresponds to a specific energy level, and the number of independent eigenfunctions relates to the degeneracy of that energy level. In this case, the energy of the third excited state is (9/2) \u0127\u03c9, and there are 10 independent eigenfunctions associated with this energy level."}