{"knowledge_schema": {"broad_category": "Quantum Mechanics \u2192 Harmonic Oscillator", "refinement": "This problem pertains to the quantum harmonic oscillator, which describes a particle in a quadratic potential well, specifically in three dimensions.", "specific_scope": "The focus is on determining the energy of the third excited state and the number of linearly 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 linearly independent eigenfunctions for 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 third excited state (n=3), the energy is (9/2) \u0127\u03c9. Additionally, in three dimensions, each energy level can have multiple linearly independent eigenfunctions, specifically 3n for the n-th level. Therefore, for the third excited state, there are 10 linearly independent eigenfunctions. The correct answer is (9/2) \u0127\u03c9 for energy and 10 for the number of eigenfunctions."}