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Feb 09, 20251 min read

$⟨ (Δ x)^{2} ⟩_{\pm} = ⟨ (Δ x)^{2} ⟩_{d} \mp 2 ∣ ⟨ x ⟩_{ab} ∣^{2}$

The variance in position of a two state system for either bosons ( $+$ ) or fermions ( $-$ ) is given by the variance in position for the distinguishable particle case $\mp$ a correction factor
Net effect is attraction between bosons and repulsion between fermions

$\displaystyle {\left\langle{(\Delta x)^{2}}\right\rangle}_{\pm }={\left\langle{(\Delta x)^{2}}\right\rangle}_{d}\mp 2\lvert {\left\langle{x}\right\rangle}_{ab}\rvert^{2}$
$\displaystyle {\left\langle{x}\right\rangle}_{ab}\equiv \int x\psi_{a}(x)^{*}\psi_{b}(x) \, \mathrm{d}x$