03_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II 3–1 Instructor: Prof. Robert Field 5.74 RWF LECTURE #3 ROTATIONAL TRANSFORMATIONS AND SPHERICAL TENSOR OPERATORS Last time; 3-j for coupled uncoupled transformation of operators as well as basis states 6-j, 9-j for replacement of one intermediate angular momentum magnitude by another patterns — limiting cases — simple dynamics minimum number of control parameters needed to fit or predict the I ( ω ) or I ( t ) Today: Rotation as a way of classifying wavefunctions and operators Classification is very powerful — it allows us to exploit universal symmetry properties to reduce complex phenomena to the unique, system specific characteristics Recall finite group theory * Construct a reducible representation — a set of matrix transformations that reproduce the symmetry element multiplication table * represent the matrices by their traces: characters * reduce the representation to a sum of irreducible representations * each irreducible representation corresponds to a symmetry species (quantum number) * selection rules, projection operators, integration over symmetry coordinates * the full rotation group is an dimension example, because the dimension is , there are an infinite number of irreducible representations: the J quantum number can go from 0 to * because the symmetry is so high, most of the irreducible representations are degenerate: the M J quantum number corresponds to the 2 J +1 degenerate components. All of the tricks you probably learned in a simple point group theory are applicable to angular momentum and the full rotation group. Rotation of Coordinates two coordinate systems: XYZ fixed in space xyz attached to atom or molecule± 2 angles needed to specify orientation of z wrt Z one more angle needed to specify orientation of xy by rotation about z± EULER angles - difficult to visualize — several ways to define — you will need to invest some effort if you want a deep understanding Superficial path θφ l m θ φ = Y l m (, ) completeness θφχ R ( ,,) l = m m m R ( φθχ l l , ,) l m 4 m 1 2 44 3 444 l ) D () , (,, mm * Rotation does not change l *
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l l 5.74 RWF LECTURE #3 3–2 [Non-lecture proof: (i) [ l 2 , l ] = 0, j (ii) rotation operators have the form e i l j α (rotate by α angle about the ˆ j axis) therefore: [ l 2 , R
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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03_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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