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 Problem Statement

Its is known that: ##L^2=L_z^2+L_{}L_{+}L_z##
##L_{+}=L_x+iL_y##
##L_{}=L_xiL_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_xiL_y=L_{}##
b.##[L_{+},L_{}]=[L_x+iL_y,L_xiL_y]=(L_x+iL_y)(L_xiL_y)(L_xiL_y)(L_x+iL_y)=##
##=L_x^2iL_xL_y+iL_yL_x+L_y^2(L_x^2+iL_xL_yiL_yL_xL_y^2)##
##=L_x^2iL_xL_y+iL_yL_x+L_y^2L_x^2iL_xL_y+iL_yL_x+L_y^2##
##=iL_xL_y+iL_yL_x+L_y^2iL_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_xiL_y]=(L_x+iL_y)(L_xiL_y)(L_xiL_y)(L_x+iL_y)=##
##=L_x^2iL_xL_y+iL_yL_x+L_y^2(L_x^2+iL_xL_yiL_yL_xL_y^2)##
##=L_x^2iL_xL_y+iL_yL_x+L_y^2L_x^2iL_xL_y+iL_yL_x+L_y^2##
##=iL_xL_y+iL_yL_x+L_y^2iL_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?