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- 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?

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?