= Gaussian perturbation of a quantum harmonic oscillator
{title2=$\Delta E_1/\Delta E_0=\alpha^2/(\alpha^2+\beta^2)$}
For perturbation $V_0e^{-\beta^2x^2}$ and oscillator inverse length $\alpha=\sqrt{m\omega/\hbar}$, direct Gaussian expectation values give $\Delta E_0=V_0\alpha/(\alpha^2+\beta^2)^{1/2}$ and $\Delta E_1=V_0\alpha^3/(\alpha^2+\beta^2)^{3/2}$. A narrow perturbation samples the odd state's <node> and suppresses its shift by two extra <powers> of $\alpha/\beta$. A broad perturbation approaches a constant <energy> offset for both states.
Back to article page
