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14s_rotatangmom

# 14s_rotatangmom - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Lecture # 14 Supplement ROTATION AND ANGULAR MOMENTA The matrix elements for the rotational Hamiltonian matrix will be derived. The matrix ele- ments of several angular momentum operators will be listed, not derived, and several important commutation rules will be stated. I. Nomenclature Following is a list of quantum numbers used to label the rotational energy levels of diatomic molecules in the presence of spin and orbital angular momenta: Projection on Internuclear Axis Total angular momentum J J z = Ω = Λ + Σ Orbital angular momentum L L z = Λ Spin S S z = Σ End-over-end rotation R none, R z = 0 J N N z = L z S Projection of J on lab-fixed axis M J R + L + S (1) Electronic states are usually designated as follows: The component of the orbital angular mo- mentum along the internuclear axis, Λ, is represented by a capital Greek letter Λ = 0 1 2 3 4 Σ Π Δ Φ Γ There is a left superscript corresponding to the spin-multiplicity of the state. The spin-multiplicity is 2 S + 1, where S is the total electron spin. There is a right subscript, Ω, which corresponds to the projection of J on the internuclear axis. So an electronic state is designated 2 S +1 Λ Ω . For example 3 Δ 3 means S = 1, Λ = 2, Ω = 3. Often there is a letter in front of this symbol which tells us which such state is meant. Transitions between states are usually specified as follows, regardless of whether the transition is emission or absorption: 1
5.80 Lecture #14 Supplement Page 2 A. The upper state is listed first. B. The upper state quantum numbers are listed with a single prime, the lower state with double prime (and very rarely with no prime). C. Rotational transitions are specified as P ( J ), Q ( J ), R ( J ) (sometimes N , O , S , T also) where the letter indicates the change in the pattern-forming rotational quantum number, J in case ( a ) and N in case ( b ). | Δ J | ≤ 1. For simplicity the pattern-forming quantum number will be designated by J here. The J in parentheses is always J �� , the J of the lower level. The letter used has the meaning N O P Q R S T J = J �� 3 J �� 2 J �� 1 J �� J �� + 1 J �� + 2 J �� + 3.

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