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Is something special for a quantum system with infinite degeneracy like free particle levels? $E=\frac{\hbar^2 \vec{k}.\vec{k}}{2m}$

Edit: I mean what is physical (or mathematical) significance of infinite degeneracy?

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Knowing the degeneracy of energy levels is necessary in order to calculate many basic physical properties, such as the electronic ground state, absorption and emission spectra, etc. – Douglas B. Staple Apr 18 '13 at 16:41

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