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noblegas
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Homework Statement
Consider a particle that moves in three dimensions with wave function [tex] \varphi[/tex] . Use operator methods to show that if [tex]\varphi[/tex] has total angular momentm quantum number l=0 , then [tex]\varphi[/tex] satifies
[tex] L\varphi=0[/tex]
for all three components [tex]L_\alpha[/tex] of the total angular momentum L
Homework Equations
[tex] [L^2,L_\alpha][/tex]?
[tex] L^2=L_x^2+L_y^2+L_z^2 [/tex]?
[tex] L^2=\hbar*l(l+1)[/tex] , l=0,1/2,1,3/2,... ?
[tex] L_z=m\hbar[/tex] , m=-l, -l+1,...,l-1,l. ?
The Attempt at a Solution
[tex] [L^2,L_\alpha]=[L_x^2+L_y^2+L_z^2,L_\alpha]=[L_x^2,L_\alpha]+[L_y^2,L_\alpha]+[L_z^2,L_\alpha][/tex]. [tex][AB,C]=A[B,C]+[A,C]B[/tex]; Therefore,[tex] [L_x^2,L_\alpha]+[L_y^2,L_\alpha]+[L_z^2,L_\alpha]=L_x[L_x,L_\alpha]+[L_x,L_\alpha]L_x+L_y[L_y,L_\alpha]+[L_y,L_\alpha]L_y+L_z[L_z,L_\alpha]+[L_z,L_\alpha]L_z[/tex]. Now what?