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44 lines
1.2 KiB
Plaintext
44 lines
1.2 KiB
Plaintext
Vibrations and rotations of a diatomic molecule
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atoms vibrate about an equilibrium position r₀
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⊙~~~~~~~~~⊙
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|← r₀ →|
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(pic MISSEd) potential, with taylor approximation
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(pic MISSEd) A taylor series solution is appropriate to solve this diffEQ.
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Experiments indicate vibration (E~visible light) has more energy than rotation (E~infrared), so vibrations happen much faster.
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The Rotational eigen-value spectrum
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L²,L̂𝓍,L̂𝓎,L̂𝓏 (Hermition operators)
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Say
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L̂𝓏❙Ψ❭ = mₗħ❙Ψ❭
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(pic) mₗ is bound by λ: m has value from -λ to λ
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Introduced raising and lower operators
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L̂± = L̂𝓍 ± ιL̂𝓎 = (pic) proof = ±ħL±
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L̂𝓏(L̂ ± ❙λ,mₗ❭) => (pic) proof => L̂± ❙λ,mₗ❭ = ❙λ,mₗ±1❭
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Very important point:
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Let m=l be the maximum value of m
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L̂₊❙λ,l❭ = 0 (required because wave function goes to 0 in a forbiddin region)
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This is also true for L̂₋❙λ,l′❭ with l′ the minimum value of m
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(pic) L̂₋L̂₊❙λ,l❭ = ...
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gives λ = l(l+1)
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(pic) L̂₊L̂₋❙λ,l′❭ = ...
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gives λ = l′(l′+1)
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These show that m = -l, -l+1, 0, l
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