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> **Possible Duplicate:**  
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> [Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?](https://physics.stackexchange.com/questions/23028/proof-that-the-one-dimensional-simple-harmonic-oscillator-is-non-degenerate) 
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I'm trying to convince myself that the eigenvalues $n$ of the number operator $N=a^{\dagger}a$ for the quantum simple harmonic oscillator are non-degenerate. 

I can't see a way to do this just given the operator algebra for creation and annihilation operators. Is there an easy way to show this, or does it depend on something deeper? I'd appreciate any detailed argument or insight! Many thanks in advance.