Oscillator ground-state density in a comparison wavelength ($dP=(2\sqrt{\pi }/\lambda )e^{-4{\pi }^2{x}^2/{\lambda }^2}dx$)

A free particle whose kinetic energy equals the ground-state total energy $\hslash \omega /2$ has momentum $\hslash /a$ and de Broglie wavelength $\lambda =2\pi a$, where $a=\sqrt{\hslash /(m\omega )}$. Substitution into the normalized bound-state Gaussian yields the title density. The comparison wavelength is a useful length scale; it is not a claim that the bound state has a single free momentum or spatially uniform probability density.