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 The commutation relations for the quantum mechanical angular momentum operators are often expressed as L x L = i hbar L where the x is the vector cross product and hbar is Planck's constant divided by 2 pi and L is the operator. Setting up vector operators turned out to be complicated than I expected. I had to set up vectors of functions, rather than functions returning vectors, so I could access the components as functions. Suggestions for improvement of the code would be appreciated Note that I have a vector operator dot product function defined as well as the cross product, but that is not used in the proof. .  Position operator In:= xop = x*# &; yop = y*# &; zop = z*# &; In:= rop = {xop, yop, zop}; Momentum operator In:= pxop = -I \[HBar] D[#, x] &; pyop = -I \[HBar] D[#, y] &; pzop = -I \[HBar] D[#, z] &; In:= pop = {pxop, pyop, pzop}; Operator Dot Product Function In:= opdot[{a_, b_, c_}, {d_, e_, f_}] := (Composition[a, d][#] + Composition[b, e][#] + Composition[c, f][#]) & Operator Cross Product Function In:= opcross[{a_, b_, c_}, {d_, e_, f_}] := {(-Composition[c, e][#] + Composition[b, f][#]) &, (Composition[c, d][#] - Composition[a, f][#]) &, (-Composition[b, d][#] + Composition[a, e][#]) &} Angular Momentum operator L In:= lop = opcross[rop, pop]; L^2 In:= lsqrop = opdot[lop, lop]; L x L In:= lxlop = opcross[lop, lop]; L x L acting on an arbitary function In:= lxlopf = Through[lxlop[f[x, y, z]]] // FullSimplify; i \[HBar] L acting on an arbitrary function In:= lopf = I \[HBar]*Through[lop[f[x, y, z]]] // FullSimplify; L x L = i \[HBar] L In:= Thread[lxlopf == lopf] Out= True Attachments: