Question

ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given that the ground state energy is -13.6 eV. (b) For the wave function in part (a), determine the possible values of the total angular momentum of the electron if the electron spin is included. Hence, calculate the possible values of the magnetic moment of the atom.

Answer 1

4100 210 1 <liw! Yew> tu ? 42101 1210 > * L. 1 Hiw7 t A 140) Y(5,0) = 5 / TAR 14 (7,0)) = 4 now from normalization <4(7,091,467,0) 7 le 2 AY - 1 - 3 3 + A = 1 inte >> 7 A √ 2 / = -13.6 ev .: 415,0). To Yow that the for to Yiw => n=1 Y210 => n = 2 Emin - f En=1 -3.yev ev 13.6 22 et Ya e (3.4ev)t now y (+) + 7 ho é A Tin e 13 at + J V ta . E For average eneroy K465,9) |€140,0)) + 2/3 EZ 9 <=> [$*(13.6) +*+(-3,777er -(-4.55 - 4.99 - ?;27 )er KEYE 16.8 ev

+ 3 210 Y (5,0) - Jess You Comparing for Yew => 2 = 0 Ymim! for for 1210 => 2-2 2. nie L = (?;+z/ to 11 – 12) 1 momentum Now for Spin angular Ş, = ² / u S {.:/ 1,0 momen fun for .L=1, S. 1' : Total angular momentan 2 1:0 j=2,. S=0 For J = 1 L =) moment oc - J = 2,1,0 of the afon Magnetic u = gju where J(2+) +(St) -1 (t + 1) 1 + 21 (1+1 for J = 2 L=1) SE] 2x3 + 1x2 - 1x2 9 = 1+ 2 x 2 x 3 1+ Ź Ž: 2 for sa 1 J. = 1x2 + 1*3-lxs 2x2 + L-1) g 1+ B/L for J = 1, L=1, 8 = 1 1.x2 to-IXL + 2XL

total. anga.com momentan we For 7=2 . and 1 get 24 1./T. bohr. magneton 9.2 74.x10 now My => In J=2 3 u = ( 2 x 1977 x2 x 9.274x20?4 u = 27.8x1024111 Fr. J=1 M wla x + x 9.274x1024)3/ 13.9x1024 1/7] M Į