4) Show that the uncertainty relation given in the third question holds for the state given as Ф(Ө, Ф) %3D '(0,4) – 3) a) Show that the expectation values (L,) and (L,) are zero for an eigenstate Y" of Lz, where Y" is a spherical harmonic of degree l and order m. (Hint: Express Lx and L, in terms of the ladder operators L4, where L4Y" = JI(I + 1) – m(m± 1)Y"±1). b) Find the uncertainties ALx and AL, in the state Y, and verify the uncertainty relation given by AL,ALy 25|(L,)|- c) Show that for the eigenstates of L² and Î2, |(ALx)² + (AL,)ʻ| is minimal when m = ±l.

4) Show that the uncertainty relation given in the third question holds for the state given as Ф(Ө, Ф) %3D '(0,4) – 3) a) Show that the expectation values (L,) and (L,) are zero for an eigenstate Y" of Lz, where Y" is a spherical harmonic of degree l and order m. (Hint: Express Lx and L, in terms of the ladder operators L4, where L4Y" = JI(I + 1) – m(m± 1)Y"±1). b) Find the uncertainties ALx and AL, in the state Y, and verify the uncertainty relation given by AL,ALy 25|(L,)|- c) Show that for the eigenstates of L² and Î2, |(ALx)² + (AL,)ʻ| is minimal when m = ±l.