5.74 RWF LECTURE #3 ROTATIONAL TRANSFORMATIONS AND SPHERICAL TENSOR OPERATORS

Transcription

1 MIT Department of Chemistry 5.74, Spring 004: Introductory Quantum Mechanics II Instructor: Prof. Robert Fied RWF LECTURE #3 ROTATIONAL TRANSFORMATIONS AND SPHERICAL TENSOR OPERATORS Last time; 3-j for couped uncouped transformation of operators as we as basis states 6-j, 9-j for repacement of one intermediate anguar momentum magnitude by another patterns imiting cases simpe dynamics minimum number of contro parameters needed to fit or predict the I(ω) or I(t) Today: Rotation as a way of cassifying waefunctions and operators Cassification is ery powerfu it aows us to expoit uniersa syetry properties to reduce compex phenomena to the unique, system specific characteristics Reca finite group theory * Construct a reducibe representation a set of matrix transformations that reproduce the syetry eement mutipication tabe * represent the matrices by their traces: characters * reduce the representation to a sum of irreducibe representations * each irreducibe representation corresponds to a syetry species (quantum number) * seection rues, projection operators, integration oer syetry coordinates * the fu rotation group is an dimension exampe, because the dimension is, there are an infinite number of irreducibe representations: the J quantum number can go from 0 to * because the syetry is so high, most of the irreducibe representations are degenerate: the M J quantum number corresponds to the J+1 degenerate components. A of the trics you probaby earned in a simpe point group theory are appicabe to anguar momentum and the fu rotation group. Rotation of Coordinates two coordinate systems: XYZ fixed in space xyz attached to atom or moecue anges needed to specify orientation of z wrt Z one more ange needed to specify orientation of xy by rotation about z EULER anges - difficut to isuaize seera ways to define you wi need to inest some effort if you want a deep understanding Superficia path θφ m = Y m (, θ φ ) competeness R( θφχ,, ) m = m m R(φ, θχ, ) m m D m m, ( θφχ,, ) * Rotation does not change *

2 5.74 RWF LECTURE #3 3 [Non-ecture proof: (i) [, ] = 0, j (ii) rotation operators hae the form e i j α (rotate by α ange about the ĵ axis) therefore: [,R] = 0 no change in under rotation.] D (,, ) φθχ is a ( +1) ( +1) square matrix that specifies how the R(φ,θ,χ) rotation transforms m. Wigner Rotation matrix: D (,, φθχ ) m φθχ,, m this operates to eft (both bra and operator are compex conjugated) = R = m exp( i φ )exp( i θ )exp( i χ ) m operates to right z y z 3 successie rotations of m in order χ,θ,φ ( ) reduced rot. matrix = exp iφm i χ m d ( θ) first d t 1 [( + m )!( m )!( + m)!( m)!] () θ = ( 1) ( + m t)!( m t )! t!( t+ m m )! t t θ +m m θ cos sin t m +n t ranges oer a integer aues where the arguments of the factorias are not negatie. d is rea and has ots of usefu properties So now we now how to write the transformation under rotation of any anguar momentum basis state.,m are abes of an irreducibe representation of the fu rotation group. Here is the wonderfu part: the rotation matrices actuay hae the form of anguar momentum basis functions! +1 1 integer ( ) D (φ,, θχ ) = θ φ 4π Y m (, )* (anguar part of atomic orbita) m0 χ is irreeant except for phase choice ( J ) D mj (φ,, θχ) = 8π [ 1 (J +1)] φθχjmk,, * sym. top waefunction

3 5.74 RWF LECTURE #3 3 3 Not coered in Lecture Suppose we hae a matrix eement of some operator A JM A J M = A JMJ M This is a number. Rotate a 3 terms in the matrix eement [ JM R 1 ]RAR 1 [R J M ] = A JMJ M If A can be expanded as a sum of terms that transform under rotation as anguar momentum basis functions, then the integra is a sum of products of D(φ,θ,χ) matrices. But the integra cannot depend on the specific aues of φ, θ, χ. This tes us that, if we can partition A into a sum of terms, each of which has the rotationa transformation properties of an anguar momentum basis state, we wi be abe to eauate the anguar part of the integra impied by the matrix eement. This is actuay how the Wigner-Ecart Theorem is deried. Spherica tensor expansion is ie a mutipoe expansion. Anything can be broen up into anguar momentum-ie parts, incuding what a aser writes onto a moecuar sampe. A(r,θφ), = a J () r JM, Y JM spherica tensor operators T q (, θφ) (ie the anguar, radia separation of the hydrogen atom waefunction) a spherica tensor or ran is a coection of + 1 operators that transform among each other under rotation as = m = q Wigner-Ecart Theorem! J J NJM T q N J M = ( 1 ) J M NJ T N J M q M proportionaity constant: Notation: T q ( A) cassification is wrt specific anguar momentum!! reduced matrix eement x,y,z are ector wrt L, J but not S, etc. means some combination of the components of {A} that satisfies the coutation rues z, q q [J T (A)] = qt (A) [J±, T (A)] = [ +1) q q ±1)] 1 T q ( ( q±1 (A) Aternatiey, thin of defining projection operators that project out of an arbitrary operator, syetryabeed operators.

4 5.74 RWF LECTURE #3 3 4 If A is a ector, ie r = xi ˆ + yj ˆ () T 1 0 ( A) = A z + z ˆ or L or S or J 1 T () ±1 ( A) = m 1 (A X ± ia Y ) (not the same as A ± ) xx xy xz yx yy yz if A is a second ran Cartesian tensor, ie zx zy zz T 0 0 ( A) = 3 1 (xx + yy + zz) nine components () 1 T 1 0 ( A ) = i (xy yx) e. g. (L xl y L y L x = ihl z) ( T 1) ± 1 ( A ) i L x L {±L + il y} =± h y x T 0 ( A) = T ±1 = mi {(yz zy) ± i(zx xz)} e.g. ± {(ih ) ± i(ih )} = h L ± 1 6 {zz xx yy} 1 ( A ) = m {(xz zx) ± i(yz + zy)} 1 ( A ) = {(xx yy) ± i(xy + yx)} T ± it is aso possibe to construct a 3 3 = 9 dimensiona reducibe spherica tensor out of two different ectors 3 x 3 = 9 reduces to RANK: (0) (1) () 1 u = 3 (u 1 1 u 0 T 0 0 (, ) 0 + u 1 1) = 3 1 (u + u + u () 1 x x y y z z) T 1 u (u u 1 1) = 0 (,) = (u () 1 (, ) =± 1 T ±1 u scaar product 1 x y u y x ) ±1) ector cross product (u± 1 0 u 0 1 x z z x ) m i(u u = [(u u y z z y)] 1 u = 6 (u u 0 T 0 (, ) ( ) (, ) T ±1 u 0 + u 1 1) = 6 1 (u u u = 1 (u + u = m 1 T 1 ± (, u ) = u ±1 ±1 = [ u z z x x y y) ±1) ±1 0 0 [ u x z + u z x ± i(u y z + u z y)] u ± i (u + u x x y y x y y x )]

5 5.74 RWF LECTURE #3 3 5 Transformations between SPACE and moecue fixed coordinate systems A rotationa waefunction specifies the probabiity ampitude distribution of orientation of the moecue fixed axis system in the aboratory axis system. Origin of both coordinate systems is fixed at moecuar center of mass. We can express the reationship between LAB and moecue frame quantities. For exampe, radiation is specified in the LAB, but transition moments are tied to the moecue frame. The moecue radiation interaction ε µ is a dot product between quantities specified in different coordinate systems. We must transform one of these into the coordinate system of the other. direction cosines P ω q T ( A ) = D Pq * T A LAB q moecue ω = (φ,θ,χ) a 3 Euer anges inerse: ( ) T ( A ) = D Pq ( ω )TP ( A) q D Pq P P, q =D ( ω) * we now this from the form of D exp( ijzφ)d(θ) exp( ijzχ) syetric top waefunction J =, M= P, Ω = -q ω JΩM = [(J + 1)8π ] 1 D MΩ (ω)* ( Thus the action of Tq A) on JΩM is an exampe of an uncouped to couped transformation and a matrix eement of D is two such transformations J MD ( Ω ) ( ω ) * J Ω M = ( 1) M Ω [(J + 1 )( J + 1)] Pq J J J J Ω q Ω M P M moecue LAB (the order of the coumns is rearranged from what we expect from ector couping, but compatibe with syetry properties of 3 j) It is in form of W-E Theorem 1

6 5.74 RWF LECTURE #3 3 6 These are the direction cosine matrix eements! For a aboratory frame ε(t) fied poarized aong the z-axis H = T 1 (ε) T ( µ ) t P= 0 t q 0 1 t for a transition poarized aong the moecue z-axis, H( t ) = ε Z () td 1 00 ( ω )* µ z ω q µ) = T (ε) D 0q * T (µ) 0 q= 1 rotate moecue frame into space frame JΩ M J H( t) J Ω M = ε z ()µ ( 1) J M J J t z ( 1) J 1 J J 1 J M 0 M Ω 0 Ω J J J Ω P ( M seection rue) q ( Ω seection rue) 1 [(J + 1 )( J + 1)]. This resut is based on the spherica harmonic addition theorem which is deried in A. R. Edmonds Anguar Momentum in Quantum Mechanics, page 6: π π π ( j ( j ) D j ) (αβγ)dm m (αβγ)d 3 (αβγ)dα sinβ d β dγ j j 3 j 1 j = π 1 j j 3 8 m 1 m m m 1 m 3 3 m