Question

Quantum Mechanics:

(Angular momentum) Write down the Hamiltonian of a rigid body in terms of angular momentum operators and the principal moments of inertia. Discuss the commutation relations for both space-fixed axes and body-fixed axes. Discuss the special case-spherical top, obtain the eigenvalues of the Hamiltonian for the spherical top.

Answer 1

I IZ Hamiltonian Ly Lz 2 In 2 Ty where Ix, fyĽ are angular momentum along x, y, ans respectively f principle moment of Inertia clong ni Y, ez anis rubectively Space -fided - n, y, z body fined y Lu, ly, Zz anel anu

was X Z T= VD y ba by bz Lix & + ily + Å L2 + 5 ( 2px - up) Flybz -2 by) + + Ê laby-ybal equating cofficient 2x = yh2-epy , 2y = abr - ab 2 7 22 = uby-ybu

[ly, x] = [zpre-ab , x] = [zkx, z] - [xf , x] 7 (ww] + [3,4]"" [by,60] - [bs, mibz - Z =-2 х ih [ly, an] =-it [Ly, y] = [7px - 34b2, 3] - [zbaix] - [152.x] [bi] + [20]*BX-2101,31-L/juz [ 13, 2] = [zba- ubz, z] = [7ba17] - [na, z] - x[b2, z] - [y] P2 = -xx-it [by, z] iza

Good Luck Page No Date spherical top In = Iy = Iz Hamiltonian 2 a2 f gIx + H 2 Is x 2 L + 2y + 2 ( By definition À I (13 : 12 g In wave function 14 - 1 l, m) - 17,m10, d)) Schrodinger î? Ely) - hley 11,m) lll+1) h² E14) egh 21x gIx lll+1) h²14) 2 In E elett 2 In (eignralne)