Ladder Operators: Commutation Relation & Beyond

[Gbox]

Homework Statement

Its is known that: $L^2=L_z^2+L_{-}L_{+}-L_z$
$L_{+}=L_x+iL_y$
$L_{-}=L_x-iL_y$

a. what is $L_{+}^{\dagger}$
b. what is $[L_{+},L_{-}]$
c. what is $||L_{+}|l,m>||^2$
d. assuming all coefficients are integer and positive what is $L_{+}|l,m>$

Relevant Equations

$L^2=L_x^2+L_y^2+L_z^2$

a. $L_{+}^{\dagger}=(L_x+iL_y)^{\dagger}=L_x-iL_y=L_{-}$

b.$[L_{+},L_{-}]=[L_x+iL_y,L_x-iL_y]=(L_x+iL_y)(L_x-iL_y)-(L_x-iL_y)(L_x+iL_y)=$
$=L_x^2-iL_xL_y+iL_yL_x+L_y^2-(L_x^2+iL_xL_y-iL_yL_x-L_y^2)$
$=L_x^2-iL_xL_y+iL_yL_x+L_y^2-L_x^2-iL_xL_y+iL_yL_x+L_y^2$
$=-iL_xL_y+iL_yL_x+L_y^2-iL_xL_y+iL_yL_x+L_y^2$
$=-2iL_xL_y+2iL_yL_x+2L_y^2=2(iL_xL_y+iL_yL_x+L_y^2)$

It is not ture that $L_yL_x=L_xl_y$ right? What can be done next?

 

[George Jones]

It is not ture that $L_yL_x=L_xl_y$ right?

Right. But $L_x L_y - L_y L_x =$ ?

Also, be careful with the +- signs. The $L_y^2$ terms should cancel.