Why does the quantum harmonic oscillator have discrete energy levels?

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

What happens when I substitute [tex]\psi_n[/tex] with [tex]\psi_0[/tex]?
 
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oops, sorry for the silly question...
of course I will get 0.