phy-4600/lecture_notes/3-18/overview

Vibrations and rotations of a diatomic molecule

    atoms vibrate about an equilibrium position r₀

    ⊙~~~~~~~~~⊙
    |←   r₀  →|

    (pic MISSEd) potential, with taylor approximation

    (pic MISSEd) A taylor series solution is appropriate to solve this diffEQ.

    Experiments indicate vibration (E~visible light) has more energy than rotation (E~infrared), so vibrations happen much faster.

The Rotational eigen-value spectrum

    L²,L̂𝓍,L̂𝓎,L̂𝓏 (Hermition operators)

    Say
        L̂𝓏❙Ψ❭ = mₗħ❙Ψ❭

        (pic) mₗ is bound by λ: m has value from -λ to λ


Introduced raising and lower operators

    L̂± = L̂𝓍 ± ιL̂𝓎 = (pic) proof = ±ħL±

    L̂𝓏(L̂ ± ❙λ,mₗ❭) => (pic) proof => L̂± ❙λ,mₗ❭ = ❙λ,mₗ±1❭

    Very important point:

        Let m=l be the maximum value of m
        L̂₊❙λ,l❭ = 0 (required because wave function goes to 0 in a forbiddin region)
        This is also true for L̂₋❙λ,l′❭ with l′ the minimum value of m

    (pic) L̂₋L̂₊❙λ,l❭ = ...
            gives λ = l(l+1)

    (pic) L̂₊L̂₋❙λ,l′❭ = ... 
            gives λ = l′(l′+1)

    These show that m = -l, -l+1, 0, l