I was watching the following video on the harmonic oscillator using ladder operators :
http://youtu.be/gRdCV9p8sAU?t=30m9s
Clicking on the video above will take you to the exact point where my questions are based off (30:09).
At that point, he has circled in black the step up and step down operators and how they act on the wavefunction $\psi$. He does not explain how he obtains the normalisation constants $\sqrt n$ and $\sqrt{n+1}$.
Here's my attempt starting with the step down operator (C is my normalisation constant) :
$$\int{(a^-\psi)^*a^-\psi}.dx=1$$ $$\int{ C^*\psi^*_{n-1}a^-\psi}.dx=1$$ $$\int{C^*\psi^*_{n-1}C\psi_{n-1}}.dx=1$$ $$\mid C\mid^2\int{\psi^*_{n-1}\psi_{n-1}}.dx=1$$
Where do I go from here ?
EDIT : I would also like to know whether the equation for the energy of the harmonic oscillator is derived from pure observation of pattern or is there a general method to derive it ?