# Proof of ladder operator identity

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1. Dec 13, 2015

### shinobi20

1. The problem statement, all variables and given/known data
Define n=(x + iy)/(2)½L and ñ=(x - iy)/(2)½L.

Also, ∂n = L(∂x - i ∂y)/(2)½ and ∂ñ = L(∂x + i ∂y)/(2)½.

with ∂n=∂/∂n, ∂x=∂/∂x, ∂y=∂/∂y, and L being the magnetic length.

Show that a=(1/2)ñ+∂n and a=(1/2)n -∂ñ

a and a are the lowering and raising operators of quantum mechanics.
2. Relevant equations

3. The attempt at a solution

Sorry but I really don't have any idea on how start. I just know that a=(1/ (2)½) (x/x0-ip/p0) and a=(1/ (2)½) (x/x0+ip/p0) with x0=(ħ/mω)½ and p0=(ħmω)½

2. Dec 13, 2015

### nrqed

Have you tried calculating the commutator of your a and $a^\dagger$? The idea is to show that they are the same as for the raising and lowering operators.

3. Dec 14, 2015

### shinobi20

Wouldn't that just be verifying but not showing? I think the problem wants me to derive it.