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I understand why when the ladder operators act upon an energy eigenstate of energy E it produces another eigenstate of energy E [tex]\mp\hbar \omega[/tex]. What I don't understand is why the following is true:

[tex]\ a \left| \psi _n \right\rangle &= \sqrt{n} \left| \psi _{n-1} \right\rangle [/tex]

[tex]\ a^{\dagger} \left| \psi _n \right\rangle &= \sqrt{n+1} \left| \psi _{n+1} \right\rangle [/tex]

I don't really even know what [tex] \left| \psi _n \right\rangle[/tex] represents, though I think it is something to do with the state of a system. How can you derive the above property?

I am a bit of a beginner to SHO in quantum theory.

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# Ladder Operators acting upon N Ket

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