​ (a) State all the energy levels of a symmetric top with principal moments of inertia I1 = I2 = I I3. ​

Question

(a) State all the energy levels of a symmetric top with principal moments of inertia I₁ = I₂ = I \(\ne\) I₃.

(b) A slightly asymmetric top has no two I's exactly equal, but I₁ - I₂ = A \(\ne\) 0, I₁ + I₂ = 2I, (A/21) << 1. Compute the J = 0 and J = 1 energies through O(A).

Answer 1

(a) Let (x, y, z) denote the rotating coordinates fixed in the top. The Hamiltonian of the system is

Hence a state with quantum numbers J, m has energy

which gives the energy levels of the symmetric top.

(b) For the slightly asymmetric top the Hamiltonian is

Hence for the perturbed states:

(i) J = 0, m = 0, (nondegenerate):

(ii) J = 1, m = 0, (nondegenerate):

(iii) J = 1, m = fl, (two-fold degenerate):

As degeneracy occurs, we first calculate

which means that the energy of the states J = 1, m = h, E_{\(1,\pm 1\)} splits into two levels: