= Nondegeneracy propagated by an oscillator ladder
{title2=$\dim\ker(H-E_n)=1\quad\text{if}\quad\dim\ker(H-E_0)=1$}
For the <quantum harmonic oscillator> with $a_-=P-im\omega X$, $a_+=a_-^*$ and $\gamma=2m\hbar\omega$, factorization gives $a_+a_-=2m(H-E_0)$, where $E_0=\hbar\omega/2$. For an <energy eigenstate> $\psi$ at $E_n=E_0+n\hbar\omega$,
$$
\|(a_-)^n\psi\|^2=\gamma^n n!\|\psi\|^2,\qquad (a_+)^n(a_-)^n\psi=\gamma^n n!\psi.
$$
The first identity shows that lowering to the <ground state> is injective on the $E_n$ <eigenspace>; the second shows how to invert it by raising. If the ground <eigenspace> is one-dimensional, $(a_-)^n\psi=c\psi_0$ and therefore $\psi=c(a_+)^n\psi_0/(\gamma^n n!)$. Thus every level is also nondegenerate. This proof explicitly rules out a second independent <energy eigenstate> at an excited level rather than merely constructing one ladder of states.
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