The energy eigenfunction of an electron in a h atom is iy - 2z) exp(-a/x2 +y2 + z2 {constant(r real a> 0 determine (a) the energy eigenvalue, and (b) the probabilities that measurements of l2 and l2 yield the results 2h2 and 0 respectively. hint: use the data on page 6. [6] useful data and formulas (continued) r rsinicos o+ j sino) r cos k spherical coordinates dr = rdr êardeer sin odo rh orthonormal set f, eo, e 1ð 1 1 sin .2 sin2 0 do2 гдр2 ladder operators j= j tij2 for angular momentum j: 4v hv/(i m) (jt m +1) mt1 li = he=i¢ i cot operators for orbital angular momentum: la = - 2t d cos 0 yem 2 = 1: spherical harmonics yem(0,) normalised to dd 3 sin 0 e-i v87t 3 y10 == 4t - sin 0 e yoo = ; y = - v4t yi.-1 = cos , 15 sin2 0e2id 15 y22 32 y21 = - 8 1(3cos 0 1) cos 0 sin 0 eto y20 = 15 y2-1= 15 sin2 e-2i v 32 y2-2 = v8t (e+ m)! eimo лв7 v(e-m)! sin" 0 -m 20+ 1 (1) yem(0,) = d sin26 d cos 47 e(r)2 = 1: hydrogenic radial wave functions rnt(r) nomalised to 3/2 zr ri0(r) 2 еxp ao 3/2 z zr z r20(r exp 2an 2ao 2aa 3/2 zr zr xt aoy3 2a0 2a0 3/2 2zr zr r30(r) = exp 3a 3ao 3ao an 3/2 4/2 zr ( zr zr exp = зао 6a 3/2 2/2 (zr 27/5 2 z zi r32(r) еxt 3a0 3a0 ( )