Doubt in the quantum harmonic oscillator

[Lebnm]

I was reviewing the harmonic oscillator with Sakurai. Using the annihilation and the creation operators $a$ and $a^{\dagger}$, and the number operator $N = a^{\dagger}a$, with $N |n \rangle = n | n \rangle$, he showed that $a | n \rangle$ is an eigenstate of $N$ with eigenvalue $n - 1$, so he concludes that $a | n \rangle \propto | n - 1 \rangle$. But, to it be true, the spectrum of $N$ should be non-degenerated, shouldn't it? Is this true? Can I proof this?

 

[Alex Petrosyan]

You already label the eigen values by their energy, which relies on that the eigenvalues are non-degenerate.

 

[Lebnm]

So, in the definition of the eigenstates of $N$, I am imposing that they are non-degenerated, right? Is there some physical reason to do this? I can't see why the spectrum of $N$, and consequently the spectrum of the hamiltonian, need to be non-degenerated in a harmonic oscillator.

 

[DrClaude]

I can't see why the spectrum of $N$, and consequently the spectrum of the hamiltonian, need to be non-degenerated in a harmonic oscillator.

You'll have to look in an introductory textbook, such as Griffiths.