Suppose a half-integer l-value, say 1/2, were allowed for orbital angular momentum. From

we may deduce, as usual,

Now try to construct Y_1/2,_-1/2(θ,ϕ) by (a) applying L_ to Y_1/2,_-1/2(θ,ϕ); and (b) using L_ to Y_1/2,_-1/2(θ,ϕ) = 0. Show that the two procedures lead to contradictory results . (This gives an argument against half-integer !-values for orbital angular momentum.)

L+Y1/2,1/2(0,0) = 0,