1. The components of the angular momentum operator are: La = до. Ly = 无 ħ a a sin o cot 0 cos i ae ħ a a cos cot & sin o

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1. The components of the angular momentum operator are: La = до. Ly = 无 ħ a a sin o cot 0 cos i ae ħ a a cos cot & sin o i ae ao ħ a і до L. Liz = (a) Show that the ladder operators for orbital angular momentum can be written: L+ = heid a a + icot ae ao a a + i cote ae L_ = he io до, = (b) Prove that [L+, L-) = 2ħL, using the expressions for the ladder operators given in part (a). (c) Use the explicit form of the ladder operators given in (a) with the expressions for the spherical harmonics given on page 56 of the notes to show explicitly the following LUY V2ħY,1 L+Y+1 = 0 V2nyi = LY1 =