Relativistic oscillator ground-state kinetic correction ($\Delta E_0=-3{\hslash }^2{\omega }^2/(32m{c}^2)$)

The expansion of the relativistic kinetic energy at fixed rest mass gives the perturbation $-p^4/(8m^3{c}^2)$. In the harmonic-oscillator ground state, $⟨p^4⟩=3(m\hslash \omega /2{)}^2$, proving the displayed first-order energy correction. Its small parameter is $\hslash \omega /(m{c}^2)$. This correction lowers the nonrelativistic energy and does not include other relativistic interactions.