Consider a 1D phase space whose generic points are denoted as $(x,p)$. We know that the Weyl quantization $\mathrm{Op}\left(\frac{x^2+p^2}{2}\right)$ is the harmonic oscillator Hamiltonian, whose spectrum is $\left\{n+\frac12:n\in\mathbb Z_{\geq 0}\right\}$.
Question: What is the spectrum of the "square root" of the harmonic oscillator, i.e., $\mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}}\right)$? (Is it even well-defined?) Is it $$\left\{ \sqrt{n+\frac12}:n\in\mathbb Z_{\geq 0}\right\},$$ or not?
My caution is that it may not be true that $$\mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}}\right)^2 = \mathrm{Op}\left(\frac{x^2+p^2}{2}\right),$$ but instead $$\mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}}\right)^2 = \mathrm{Op}\left(\sqrt{\frac{x^2+p^2}{2}} \star \sqrt{\frac{x^2+p^2}{2}}\right),$$ where $\star$ is the Moyal star operator. I don't know whether this Moyal product equals to $\frac{x^2+p^2}{2}$.
Please note that I am a physicist, and I only have bare knowledge on pseudodifferential operators, etc. However, any help is appreciated!