= Relative-coordinate moments in even and odd oscillator ground states
{title2=$\mathbb E_{\mathrm F}[y^n]=(n+1)\mathbb E_{\mathrm B}[y^n]$}
The lowest even <quantum harmonic oscillator> <quantum state> has density $\rho(y)=e^{-y^2/\tau^2}/(\sqrt\pi\tau)$, and the lowest odd <quantum state> has density $2y^2\rho(y)/\tau^2$. Since $\rho'=-2y\rho/\tau^2$, <integration by parts> gives $\mathbb E_{\mathrm F}[f]=\mathbb E_{\mathrm B}[f+yf']$ whenever the boundary terms vanish. For $f(y)=y^n$, this proves the displayed identity. Odd moments vanish in both <quantum states> by <parity>; even moments quantify the larger spatial spread in the odd <quantum state>. In particular the relative-coordinate <variance> is three times as large, and the odd density vanishes at particle coincidence.
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