How to Derive Raising and Lowering Operators from Ladder Operator Definitions?

[shinobi20]

Homework Statement

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.

Homework Equations

3. The Attempt at a Solution [/B]
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

 

[nrqed]

Homework Statement

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.

Homework Equations

3. The Attempt at a Solution [/B]
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

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.

 

[shinobi20]

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.

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