Question

Problem 3 The Hamiltonian of a rotator is given by where 11 and 13 are moments of inertia, and Ly, Ly, and L, are the compo- nents of the orbital angular momentum operator. 1. Determine the eigenvalues of the Hamiltonian and their degeneracy in the two limits 11 = 13 and 11 > 13. 2. Sketch the energy spectrum in these two limits. 3. What is the energy spectrum in the limit 11 > 13? Problem 4 Consider the hermitian matrix -3 19/4e**/3 (19/4e-ix/3 6 ) 1. Find the eigenvalues and corresponding eigenvectors. 2. Find the matrix U which diagonalizes A. 3. Calculate the matrix exp(-A).

Answer 1

1-3 Jig ens - Well 6 .. to Super values 1A-211 =0 1-3-7 Jig eins var eine Geo 364 7 +(+8)(3+2) – 19. Do 7 2=64+39 –18 - 1430 422- 249 -72–19 20 122–122 - 91.30 7:422-269 +49-84 20 1.9 2(22+7) – 13 (22 + 7) =0 7 (22–13) (22+7)=0 So, Eisenialus. 4X7X13 2727,13

Osigen voeltors for = 1242 out 3) =0 > -19 x + 1 ta ei sy zo A 19 anysx - Ysy 20.' So; Esen vector, a o · © Ligon-veetos foros dz -, ==Iя е So, Eisen rectes,

Stithe Siagonal Mätning The Modal Matrix 1. Ti Trigerants T1+19 e av varors Vige117) 4 190 V Vae1%) i VT +1246) +377) Svie ēns Viges) MF 1358 liaeth 1 ) diagonalised the hermitian So This matrix Matrix A. . AMA = Diagonal Matrix * exp(t) =* = 1-A71 A +- A = (-3 m ) Az X-3 Heim /-3 qer a 6 i 6)