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Ladder Operators

  • Thread starter Gbox
  • Start date
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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_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

Staff Emeritus
Science Advisor
Gold Member
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737
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.
 

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