How can [a+,[a+,a]]=0 be proven in the quantum oscillator system?

[meanyack]

Homework Statement

Actually the question is two long and I'll be done if I can show that
[a⁺,[a⁺, a]=0 and similarly
[a,[a⁺, a]=0
where a⁺ is the raising and a is the lowering ladder operator in quantum oscillator.

Homework Equations

I tried the formulas
[A,[B,C]]= -[C,[A,B]] -[B,[C,A]] and
a$$\psi$$_n=$$\sqrt{n}$$$$\psi$$_n-1
a⁺$$\psi$$_n=$$\sqrt{n+1}$$$$\psi$$_n+1

The Attempt at a Solution

 

[keniwas]

I tried the formulas
[A,[B,C]]= -[C,[A,B]] -[B,[C,A]] and

The Jacobi Identity is [A,[B,C]] = [[A,B],C] + [B,[A,C]]

 

[meanyack]

why can't I see a button "delete topic" because this is the wrong topic, the original one is the other one