Question

a homonuclear diatomic with nulcear masses and intrinsic spins $I_1=I_2$ . the molecule is in its electronic ground state and behave like a rigid rotor. The Hamilitonion for this mition

M1 = M2

$H(\theta ,\phi )=-\frac{h}{2\mu r^2}(\frac{1}{\sin \theta }\displaystyle \frac{\partial }{\partial \theta }\sin \theta \frac{\partial }{\partial \theta }+\frac{1}{sin^2\theta }\frac{\partial^2 }{\partial \phi^2 })$

Find the Eigenvalues and Eigenfunction to this nuclear moition, and degenraices

Answer 1

Quantum Mechanics Problem Upamanyu M January 5, 2018 Given the Hamiltonian, We use the separation of varibles method to find the eigenfunctions ψ(θ, φ) and the associated eiges (we have normalized the eigenvalue so that E is dimensionless and ur2 has the dimension of energy) Let us write: We therefore have sin Or sin θ d (ΙΘ sin' θ constant Now we would have single-valued solution for Φ only if, with m integer, so that Φ ~ exp(timo) Now, putting this back into equation (5), we have for θ, As is well-known from the theory of differential equation, this differential equation has regular solution everywhere only when

pamanyu M Chegg Solution where 1 is a non-negative integer and the values of m run from -1 to +l (2l +1 values of m). The eigenfunctions are the associated Legendre polynomials (lml Therefore, the eigenvalue of the Hamiltonian is: 1) Θ(θ)=pr (cos θ) 120) (10) The eigenfunction is (12) COS where A is a normalization constant. As such, this is (2l + 1)-fold degenerate (the eigenvalue is same for each value of m l). But there is another source of degeneracy, the nuclear spin. Each of the nuclei can exist in any of 21 + 1 spin states. So the total degree of degeneracy is: d-(2l +1)(21 1) (212 +1) (13)