Why does the quantum harmonic oscillator have discrete energy levels?

[glederfein]

Hello.
I am trying to use the following equation:
$$a\left|\psi_n\right\rangle=\sqrt{n}\left|\psi_{n-1}\right\rangle$$
(where a is the "ladder operator").

What happens when I substitute $$\psi_n$$ with $$\psi_0$$?

 

[glederfein]

oops, sorry for the silly question...
of course I will get 0.