{"knowledge_schema": {"broad_category": "Physics \u2192 Quantum Mechanics \u2192 Harmonic Oscillator", "refinement": "This problem pertains to the quantum harmonic oscillator, which describes a particle in a quadratic potential well.", "specific_scope": "The focus is on determining the energy levels and the degeneracy of the second excited state of a three-dimensional isotropic harmonic oscillator.", "goal": "Calculate the energy of the second excited state and identify the number of linearly independent eigenfunctions corresponding to that energy."}, "summary": "In this quantum mechanical problem, we analyze a particle in a three-dimensional isotropic harmonic oscillator potential. The energy levels of such a system are quantized and given by the formula E_n = (n + 3/2) \u0127\u03c9, where n is the quantum number. For the second excited state (n=2), the energy is (7/2) \u0127\u03c9. Additionally, the degeneracy of this state, which indicates the number of linearly independent eigenfunctions, is determined by the number of ways to distribute the quanta of energy among the three dimensions, resulting in 6 independent eigenfunctions. Thus, the answer is (7/2) \u0127\u03c9, 6."}