1 (c) The z-component of the orbital angular momentum I, = (aộ, – įpa), has c = x ħ the following commutation relations

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1 (c) The z-component of the orbital angular momentum I, = (aộ, – įpa), has c = x ħ the following commutation relations with the position operators: = = = 2 7 = 4 2 [2] [Î2, ] = iŷ [În, ĝ] = -iî, [Î2, ] = 0. (1) i. Using Eq. (1), find the commutator [În, frº], where t’ = ĉ2 + Û2 + 22. ii. Using the result from i, state the value of the commutator [În, f(f)], where f is a function, giving reasons for your answer. a (d) In the coordinate representation, in =-i- where ø is the azimuthal angle ao of the spherical coordinates that describes rotation about the z-axis. i. By considering the action of În on a function of o, e.g., 0(6), find the eigenvalues and normalised eigenstate wavefunctions of this operator. [Note: when Î, acts on a function of o, 3/80 = d/do.] ii. Derive a condition that must be satisfied by the functions 0;(0), for the operator În to be self-adjoint, i.e., for the following to hold: ? [5 > 4 2 27 2л 1. * [1:02(m)]*-(6)do = 1 * *3(0)[.01(%)]dø.