Question

322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h. (d) What is the probability of finding the particle at the position 0 = x/3 and p = x/2 within d0 = 0.03 rad and do = 0.03 rad?

Answer 1

AML a Since Yoo (.9,9 ) = 1 r (32%)/or and Y, , (x,y,z) =- FU 15/8 (**») pour 1. 32" h V22 22 hence and e V (Yeni - You); ( ), 1) - TT , Vàếp , -. de =ts Yo +v( 92,-,-Yer) the Having expressed in in terms of soherical harmonics, we can now easily. [ 4 12,4, 8) = 1 [Pro + V3 2 (2,1 %) = 6t of (x, y, z) Î, Y (1,3,7) = 1 l ho + V 12 (Y 21 Yes) ta ņ (%2m +/ ) This shown that 4 (X,Y, 4) is an eigenstate if aw with eigenvalue où; 4(4,92) is however not an eigenstate of 22. Thus the total angular momention of the particle is 124 12 745 6 ħ and

b Uning the relation 1 Lt Yem = t veld11), m/mt1) Veonar, we have Lt 4(x7,9) - Li Yro + Vi î, (Y,,You) atve van t ti V (vo You 2% honce (ap 191 14) ELLE <2,014 VE (<2,41 -42,1137 * [ { , i + Về (Vũ Y - 07 e. Since 147 V + V Mariel a calutation of 281 [₂ t) geelcht Lylîz 14 zo with probability Ports 24 h 1W7 = t with probability. Pop= 8/3 4 11 14) ah with probability P, 22 ¿ since 4(^, Y, 2) = -- (2.2 = xx yu) 21 mm an co-ordinate + x 2 can be written in terms of the spherical *(0,4) = n1360860-1) + Va song cooking the probability of finding the particle ent the position of and a in

P(8.4) = 14 (0,4))" sino do de :-[ (36009-1) + V3 sino lords cono Singao do hence † (16:9) (3 comment ) +0]"<0.05) in . I 9,7 x 1077