2. ( /34 pts) Consider an electron in the hydrogen atom described by the wave function 42m. (a) (6 pts) Determine the radial wave function as a function of r Rul(r) = =p Ipitle Pf (p ), using the recursion formula 2( j + 1+1-n) G+1 = (j + 1) (j + 21 + 2) ? where f (p) = Ecipi. You may use the normalization constant co = V1/5a. (b) (4 pts) Show that /42-2 is orthogonal to 420. For the next four problems, you will be considering the electron in state described by the wave function /42-2. (c) (4 pts) What are the values of n, l, m, and max? (d) (8 pts) Apply the raising operator L+ in spherical coordinates to this wave function. Use Am = hvi(l+1) -m(m +1) = hy(1 Fm)(lm+1) to normalize. Verify that the result is what you would expect. (e) (8 pts) Apply the angular momentum operator L' in spherical coordinates to this wave function. Explain the result and discuss whether it is what you would expect. (f) (4 pts) Apply the angular momentum operator Lz in spherical coordinates to this wave function. Explain the result and discuss whether it is what you would expect For reference: 1 15 Y2 2 (0, 8) = AV 27 . e-2id . sin2 0 Y 2 ' ( 0 , 8 ) = 1 15 2 V 27 . e -id . sin 0 . cos 0 Yo (0, 8 ) = AV T 1 5 . (3 cos2 0 - 1) Y 2 ( 0, 4 ) = - 1 15 2 V 27 . elo . sin 0 . cos 0 1 /15 Y2(0, 0 ) = AV 2id . sin2 0