Question

Quantum Mechanics Tutorial: Introduction to Rotational Spectroscopy

Answer 1

Note: This is a reference for educational/studying purposes, not a question, please save all comments or questions for the end.

Answer 2

\({\bf{Energy~Levels~of~Diatomic~Molecule}}\) can be calculated by assuming the molecule takes a rigid-rotator molecule where bond length is fixed

basic equations:

velocity of each molecule = 2(π)ℓv
kinetic energy = (1/2)I\(ω^{2}\)
moment of inertia = I = μ\(l^{2}\)
angular momentum = L = Iω
hamiltonian = Ĥ = \(-\frac{ ħ^{2} }{ 2μ }∇^{2}\)

Answer 3

Ĥ\[=-\frac{ ħ^{2} }{ 2μ }(\frac{ 1 }{ l^{2} \sin(θ) } \frac{ ∂ }{ ∂θ } (\frac{ \sinθ }{ ∂θ } \frac{ ∂ }{ ∂θ } )+\frac{ 1 }{ l^{2} \sin^{2}(θ) } \frac{ ∂^{2} }{ ∂φ }) \] =EY(θ,φ)
letting a new constant β = 2IE/ħ^2 and multiplying the equation by \(sin^{2}(θ)\) gives us:
\[sinθ \frac{ ∂ }{ ∂θ } (\frac{ \sinθ }{ ∂θ } \frac{ ∂Y }{ ∂θ } )+ \frac{ ∂^{2}Y }{ ∂φ^{2} } + βsin^{2}(θ)Y=0\]
separating the spherical coordinate function Y(θ,φ) into its component variables, dividing by Θ and Φ gives us
\[ \frac{ sinθ }{ Θ } \frac{ ∂ }{ ∂θ } (sinθ(\frac{dΘ}{d}) + βsin^{2}(θ)+\frac{1}{Φ}\frac{d^{2}Φ}{dφ^{2}}=0\]

treating the third term as a separation of variables problem, we get:

\[\frac{1}{Φ}\frac{d^{2}Φ}{dφ^{2}}=-m^{2}\]

this is a second-order PDE which results in φ(φ) = Ae^(imφ) which then normalizes to φ(φ) = \(\frac{ 1 }{ \sqrt{2\pi} }\)e^(imφ)

Answer 4

β = J(J+1)

therefore

E = (ħ^2/2I)(J)(J+!) = BJ(J+!) where B (not beta, capital B) is the rotational constant

degeneracy g = 2J+1

Answer 5

β = J(J+1)

therefore

E = (ħ^2/2I)(J)(J+!) = BJ(J+1) where B (not beta, capital B) is the rotational constant

- degeneracy g = 2J+1
- Selection rule states that J can only fluctuate +/-1, therefore:
- calculating the difference between E(J+1) and E(J) gives us:
Δ E = (ħ^2/2I)[(J+1 + 1)(J+1) - (J+1)(J)]
(ħ^2/2I)(2)(J+1) polynomial expansion
= (ħ^2/I)(J+1) simplification, cancelling out the 2 in the num/denom
= (h^2)/(4pi^2 I) * (J+1) converting h bar into its component form h/pi

since Δ E = hv then v = (h)/(4pi^2 I) * (J+1) or, in terms of B, 2B(J+1)

where B = h/(8pi^2 I)

or in wavenumbers

w̃ = 2B̃(J+1)