The Schr$$dinger equation for the one$-$dimensional harmonic oscillator in dimensionless variables is

$\frac{-d^2}{d{}^2}+{}^2=p$

being $p=2lo\frac{n}{127?}.$ Show that with the proposed solution $()=e^{-\frac{{}^2}{2}y()},$ we can rewrite the previous equation as

$\frac{d^{2}y}{d{}^2}-2\frac{dy}{d}+(p-1)y=0$

h$)$ Show that by comparing the previous equation with the Hermite differential equation

$\frac{d^{2}y}{d{}^2}-2\frac{dy}{d}+2ny=0$

then we obtain the quantization of the energy $lon=[127?](n+\frac{1}{2}),$ where $n=0,1,2$dots In addition, the function $y()$ is a polynomial called Hermitc's polynomial $H_n().$

c$)$ Show that the normalization constant $C_n$ for a wave function $()=C_n{e}^{-\frac{{}^2}{2}}{H}_n()$ is given by $C_n=1\sqrt[\phantom{2}]{2^n{}^{\frac{1}{2}}n!}$