MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II 3–1Instructor: Prof. Robert Field 5.74 RWF LECTURE #3ROTATIONAL 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 MJ quantum number corresponds to the 2J+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 θφlm θ φ = Ylm (, ) completeness θφχ R( ,,) l=∑m m′ m′R(φθχll , ,) lm 4m ′ 1 2 44 3 444 l θφχ )D() ′, (,,mm * Rotation does not change l*l l 5.74 RWF LECTURE #3 3–2 [Non-lecture proof: (i) [llll2, llll] = 0,j(ii) rotation operators have the form eilljα (rotate by α angle about the ˆj axis) therefore: [l2,R] = 0 — no change in l under rotation.] D()(,, )′ φθχ is a (2l +1) × (2l +1) square matrix that specifies how the R(φ,θ,χ) rotation transforms |lm〉.mmWigner Rotation matrix:D m m m i i i m mm z y z ′ = ′ℜ( ) = ′ − − − ( ) () () (,, ) ,, exp( )exp( )exp( ) l l l l l l φθχ φθχ φ θ χll ll ll operates to right 3 successive rotations of |l m〉 in order χ,θ,φ this operates to left (both bra and operator are complex conjugated) first=exp −im′−i m d ′ ( )φ χ θmm reduced rot. matrix / d() t[(l − m′)!(l − m)!]+ m′)!(l + m)!(l 12 l ′ () =∑ (−1)θmm (l −mt t t+mm′)!+m′−t)!(l− )! !( −t θ2l +mm −′−−2t θ2t m′+n ×cos2 sin 2 t ranges over all integer values where the arguments of the factorials are not negative. d()′ is real and has lots of useful propertiesmmSo now we know how to write the transformation under rotation of any angular momentum basis state. l,m are labels of an irreducible representation of the full rotation group. Here is the wonderful part: the rotation matrices actually have the form of angular momentum basis functions! / D )[χ is irrelevant exceptfor phase choice() (φθχ =2l +112 l integer θ φ,, 4π Ylm( , ) * (angular part of atomic orbital)m0 12/JDmJ ,, 82() (φθχ)=π (2J+1)] ,,φθχJMK * sym. top wavefunction5.74 RWF LECTURE #3 3–3 Not covered in Lecture Suppose we have a matrix element of some operator A JM A J M ′′ = AJMJ M ′′ This is a number. Rotate all 3 terms in the matrix element R−1 −1[R[JM ]RAR J M ′′ ′′]= AJMJ M If A can be expanded as a sum of terms that transform under rotation as angular momentum basis functions, then the integral is a sum of products of D(φ,θ,χ) matrices. But the integral cannot depend on the specific values of φ, θ, χ. This tells us that, if we can partition A into a sum of terms, each of which has the rotational transformation properties of an angular momentum basis state, we will be able to evaluate the angular part of the integral implied by the matrix element. This is actually how the Wigner-Eckart Theorem is derived. Spherical tensor expansion is like a multipole expansion. Anything can be broken up into angular momentum-like parts, including what a laser writes onto a molecular sample. A(r,θφ)= ∑aJ () θφ, r YJM (, ) (like the angular, radial separation of the hydrogen atom wavefunction) JM, () spherical tensor operators Tqka spherical tensor or rank k is a collection of 2k + 1 operators that transform among each other under rotation as |l = k m = q〉 Wigner-Eckart Theorem! J k J ′ kT()kT() ()J −MNJM N J M ′ NJ N J ′′ ′′ q =−1 −MqM′ proportionality constant:kANotation: T()()q !! “reduced matrix element” classification is wrt specific angular momentum x,y,z are vector wrt L, J but not S, etc. means some combination of the components of {A} that satisfies the commutation rules k k[JT()(A)]= qT()(A)z , q q k[JT()(A)]=[kk +1) −q q ±1)]12 T()(A)±, q ( (/ qk ±1 Alternatively, think of defining projection operators that project out of an arbitrary operator, symmetry-labeled operators.5.74 RWF LECTURE #3 3–4 r ˆ= ˆIf A is a vector, like rxi + yj + zk ˆ or L or S or J () AT01() = A z 1 T±1 A()() = m2− /12 (A ± iAY) (not the same as A ±)X xx xy xz yx yy yz if A is a second rank Cartesian tensor, like zx zy zz nine components /() AT00() =− −12 (xx + yy + zz)3 / () A −12 (xy − yx) eT01() = i2 . . (L L − L L g xy y = ihLz)x (T± 1 )() =mi / 2{(yz −zy)±i(zx −xz)} e.g. ±{(ih) ±i(ih)}= h L±1 A i Lx L {±L +iLy}=±h 2 y 2 x 2 () AT 02() = −12 {2zz − xx − yy}6 ()() = mT ±12 A 2 {(xz − zx) ± i(yz + zy)} 1 / ()() =T±22 A 2 {(xx −yy) ±i(xy + yx)} 1 it is also possible to construct a 3 × 3 = 9 dimensional reducible spherical tensor out of two different vectors 3 x 3 = 9 reduces to 1 + 3 + 5 RANK: (0) (1) (2) () −12 (u v T00(, ) = 3/ − u v 0 + u−1v1)u v 1 −1 0
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