谐振子基态波函数：
\(y_0’’=\left(\frac{m^2\omega^2}{\hbar^2}x^2-\frac{m\omega}{\hbar}\right)y_0\)
傅里叶变换：
\(y(x)=\displaystyle \int \tilde y(k)e^{ikx}dk\)
逆变换：
\(\tilde y(k)=\frac{1}{2\pi}\displaystyle \int y(x)e^{-ikx}dx\)
\(\frac{y’’}{y}=-\frac{\displaystyle \int k^2\tilde y(k)e^{ikx}dk}{\displaystyle \int \tilde y(k)e^{ikx}dk}\)
\(-\frac{y’’}{y}\)是\(k^2\)的加权平均
动量平方期望值：
\(\langle p^2\rangle=\frac{\int \hbar^2 k^2 |\tilde y(k)|^2dk}{\int |\tilde y(k)|^2dk}\)
谐振子能量\(E=p^2/2m+\frac{1}{2} m w^2 x^2\)：
\(\langle E\rangle=\frac{\hbar^2}{2m}\frac{\int k^2|\tilde y(k)|^2dk}{\int |\tilde y(k)|^2dk}+\frac12 m\omega^2\frac{\int x^2|y(x)|^2dx}{\int |y(x)|^2dx}\)
归一化：
\(\langle E\rangle=\frac{\hbar^2}{2m}\displaystyle \int k^2|\tilde y(k)|^2dk+\frac12 m\omega^2 \displaystyle \int x^2|y(x)|^2dx\)
由\(y’’(x)=-\displaystyle \int k^2\tilde y(k)e^{ikx}dk\)
经计算，最终可得：
\(\displaystyle \int y^*Ey dx=\displaystyle \int y^*\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac12 m\omega^2x^2\right)y dx\)