- #1
Another
- 104
- 5
Homework Statement
Show that ##[L_{x}^2,L_{y}^2]=[L_{y}^2,L_{z}^2]=[L_{z}^2,L_{x}^2]##
Homework Equations
##L^2 = L_{x}^2+L_{y}^2+L_{z}^2##
##L_x = yp_z-zp_y##
##L_y = zp_x-xp_z##
##L_z = xp_y-yp_x##
##[x_i,p_j]=iħδ_{ij}##
##[L_x,L_y]=iħL_z##
##[L_y,L_z]=iħL_x##
##[L_z,L_x]=iħL_y##
##[A,B]=AB-BA##
The Attempt at a Solution
##[L_{x}^2,L_{y}^2]= [L_{x}L_{x},L_{y}L_{y}]##
##[L_{x}^2,L_{y}^2]= L_{x}L_{x}L_{y}L_{y}-L_{y}L_{y}L_{x}L_{x}##
##[L_{x}^2,L_{y}^2]= L_{x}L_{x}L_{y}L_{y}-L_{y}L_{y}L_{x}L_{x}±L_{y}L_{x}L_{y}L_{x}##
##[L_{x}^2,L_{y}^2]= L_{x}L_{x}L_{y}L_{y}-L_{y}L_{x}L_{y}L_{x}+L_{y}L_{x}L_{y}L_{x}-L_{y}L_{y}L_{x}L_{x}##
##[L_{x}^2,L_{y}^2]= L_{x}L_{x}L_{y}L_{y}-L_{y}L_{x}L_{y}L_{x}+L_{y}(L_{x}L_{y}-L_{y}L_{x})L_{x}##
##[L_{x}^2,L_{y}^2]= L_{x}L_{x}L_{y}L_{y}-L_{y}L_{x}L_{y}L_{x}+L_{y}[L_{x},L_{y}]L_{x}##
##[L_{x}^2,L_{y}^2]= L_{x}L_{x}L_{y}L_{y}-L_{y}L_{x}L_{y}L_{x}+iħL_{y}L_{z}L_{x}##
##[L_{y}^2,L_{z}^2]= [L_{y}L_{y},L_{z}L_{z}]##
##[L_{y}^2,L_{z}^2]= L_{y}L_{y}L_{z}L_{z}-L_{z}L_{z}L_{y}L_{y}##
##[L_{y}^2,L_{z}^2]= L_{y}L_{y}L_{z}L_{z}±L_{z}L_{y}L_{z}L_{y}-L_{z}L_{z}L_{y}L_{y}##
##[L_{y}^2,L_{z}^2]= L_{y}L_{y}L_{z}L_{z}-L_{z}L_{y}L_{z}L_{y}+L_{z}L_{y}L_{z}L_{y}-L_{z}L_{z}L_{y}L_{y}##
##[L_{y}^2,L_{z}^2]= L_{y}L_{y}L_{z}L_{z}-L_{z}L_{y}L_{z}L_{y}+iħL_{z}L_{x}L_{y}##
##[L_{z}^2,L_{x}^2]= [L_{z}L_{z},L_{x}L_{x}]##
##[L_{z}^2,L_{x}^2]= L_{z}L_{z}L_{x}L_{x}-L_{x}L_{x}L_{z}L_{z}##
##[L_{z}^2,L_{x}^2]= L_{z}L_{z}L_{x}L_{x}±L_{x}L_{z}L_{x}L_{z}-L_{x}L_{x}L_{z}L_{z}##
##[L_{z}^2,L_{x}^2]= L_{z}L_{z}L_{x}L_{x}-L_{x}L_{z}L_{x}L_{z}+L_{x}L_{z}L_{x}L_{z}-L_{x}L_{x}L_{z}L_{z}##
##[L_{z}^2,L_{x}^2]= L_{z}L_{z}L_{x}L_{x}-L_{x}L_{z}L_{x}L_{z}+iħL_{x}L_{y}L_{z}##
It not equal. My answer is incorrect.