{"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.", "goal": "Calculate the energy of the third excited state and identify the number of linearly independent eigenfunctions for that energy eigenvalue."}, "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 can be expressed in terms of the angular frequency \u03c9. For the third excited state, the energy is given by (9/2) \u0127\u03c9, and there are 10 linearly independent eigenfunctions corresponding to this energy level. This reflects the degeneracy of the energy levels in a three-dimensional harmonic oscillator."}