In a paper, I ran into the following definition of the zero point fluctuation of our favorite toy, the harmonic oscillator: $$x_{ZPF} = \sqrt{\frac{\hbar}{2m\Omega}} $$ where m is its mass and $\Omega$ its natural frequency. However, when I try to derive it with simple arguments, I think of the equality: $$E = \frac12 \hbar\Omega=\frac12 m \Omega^2 x_{ZPF}^²$$ (using the energy eigenvalue of the $n=0$ state) giving me:
$$x_{ZPF} = \sqrt{\frac{\hbar}{m\Omega}} $$ differing from the previous one by a factor $\sqrt2$. I am just puzzled, is it a matter of conventions or is there a fundamental misconception in my (too?) naive derivation?