# Harmonic oscillator

by rayman123
Tags: harmonic oscillator
 P: 152 $$\psi_{0}= (\frac{\alpha}{\pi})^{\frac{1}{4}} e^{\frac{-y^2}{2}}$$ $$y= \sqrt{\frac{m\omega}{\hbar}}x\Rightarrow y=\sqrt{\alpha}x$$ $$\alpha= \frac{m\omega}{\hbar}$$ $$<|x^2|>=\int_{-\infty}^{\infty}dxx^2|{\psi_{0}}^2|=\sqrt{\frac{m\omega}{\pi \hbar}}\int_{-\infty}^{\infty}dxx^2e^{\frac{-m \omega x^2}{\hbar}}=I$$ $$\int_{-\infty}^{\infty}dxx^2e^{-\alpha x^2}=\frac{1}{2\alpha}\sqrt{\frac{\pi}{\alpha}}$$ $$I= \frac{1 \hbar}{2m \omega}$$ for $$<|p^2|>=\frac{m \hbar \omega}{2}$$ $$<|E_{k}| >= \frac{1}{2m}|<|p^2>|= \frac{\hbar \omega}{4}$$ $$<|E_{p}|> = \frac{m\omega^2}{2}<|x^2|>= \frac{\hbar \omega}{4}$$ can i calculate it this way? I have problems with finding formulas for the expected value for kinetic and potential energy....
 P: 152 $$<|p^2|>=-\hbar^2 \int_{-\infty}^{\infty}dxe^{\frac{-m \omega x^2}{2\hbar}}\frac{\partial ^2}{\partial x^2}e^{\frac{-m \omega x^2}{2\hbar}}\sqrt{\frac{m\omega}{\pi \hbar}}= \hbar m \omega -m^2 \omega^2<|x^2|>=\frac{m \hbar \omega}{2}$$