Question

Consider a rigid heteronuclear diatomic molecule of bond length d rotating in the xy plane about the z-axis. We have seen that the Hamiltonian for this system is H = P2/(2), where y is the reduced mass, and p = ( y) is the relative momentum. Define the angular momentum operator about the z-axis as L. = fpy - yp... a. Show that Ê, Î2] = 0. b. Show, therefore, that the eigenfunctions of A are also eigenfunctions of L. What are the corresponding eigenvalues of L,? c. Show that the Hamiltonian is given by A = 22/(21).

Answer 1

(3817). [H, La] = 17 [ (P22 +29 4 82²) , ( B - 9 in)] **u{] + [] + [6,7, bn] + [py 2, 2 ] Py + ☆ [P2² , Pg ] + [P2² , &] Py - [Perisk tor- } I as all commutators with p2 L. es zero ] The eigen function of rigid seotator Ĥ and iz is Yem (6.a) ie Yim = /21+13 (dimgi pata cred eima for 4T (fam)! for uz & eine - The corresponding eigenvalue of Lz is selet t o 0 The energy of E =45 102 rotating molecule w25 fas lysow] so, energy € = eigen value is e(lt) az ZI