ΑΜ Α and ΒΜ Β angular momentum basis states to form coupled ΑΒCΜ C basis states RWF Lecture #4. The Wigner-Eckart Theorem

Transcription

1 MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #4 4 The Wigner-Ecart Theorem It is always possible to evaluate the angular (universal) angular part of all matrix elements, leaving behind a (usually) unevaluated radial integral. Unfortunately (as in the separation of the hydrogen atom wavefunction into angular and radial factors), the radial factor depends on the values of angular momentum magnitude quantum numbers, but not angular momentum projection quantum numbers. The Wigner-Ecart Theorem provides an automatic way of evaluating the angular parts of matrix elements of many important types of operators. The radial factor is called a reduced matrix element. Often, explicit relationships between many reduced matrix elements may be derived by multiple applications of the Wigner-Ecart Theorem or by evaluating the matrix element directly for one extreme set of quantum numbers ( stretched state ) where one unique basis state in one basis is by definition identical to one unique basis state in a different basis set. The Wigner-Ecart theorem applies to systems which have lower than spherical (atoms) or cylindrical (linear molecules) symmetry. Any symmetry at all will suffice. It is essential to express all operators in spherical tensor form. It will become clear that the same operator may be expressed in several different spherical tensor forms. These are useful for evaluation of reduced matrix elements. The central idea is that operators are classified according to their transformation properties under rotation. The same transformations (Wigner rotation matrices) describe the transformation properties of angular momenta. The Wigner-Ecart Theorem may be viewed as a generalization of the coupling of separate ΑΜ Α and ΒΜ Β angular momentum basis states to form coupled ΑΒCΜ C basis states.

2 5.74 RWF Lecture #4 4 2 j m j j j njm j T q ( A ) n j m j = ( ) nj T ( A) n j m j q m j The 3-j coefficient is what you would expect for coupling j = j,m j = m j to j 2 =,m j = q 2 to form j 3 = j,m j = m j. The factor with the two pairs of vertical lines is called a reduced matrix element. It 3 does not depend on m, m j, or q! For examples (the matrix elements of the q = 0 operator component are always easiest to evaluate, especially when one chooses an m j = j basis state): T0( j) = j z (, j ) = j T j j z T0 (, j ) = 6 / (3j j ). 2 Using the W-E theorem but we also now jm j j z j m j j jm j T0( j) j m j = ( ) j T () j j, m j 0 m j j m j = δ jj δ m j m j m j. Thus, using the analytical expression for the 3-j coefficient j j 2 ( ) j m j m j( j + 2 j +)], m j 0 m j = / j [ )( we evaluate the reduced matrix element: 2 / j T () j j = δ jj [ j( j +)( 2 j +)]. Another example:

3 5.74 RWF Lecture # j m j 0 j 0 jm j T0 (, j j ) j m j = ( ) j T ( j) j m j 0 m j j 0 j = ( ) δjj δ m j m j m j 0 m j j m 0 j T ( j) j j 0 j But ( ) m j 0 m = j m (2 j + ) j 2 / thus jm j T 0 0 (, j j ) j m = δ δ m j m j (2 j + ) j jj / 2 j T 0 (, j j ) j. But, when the matrix element is evaluated directly, we have thus 2 jm j j j m j = δjj δ m j m j j( j +), 0 2 j T0 (, j j ) j = (2 j +) / j( j +). You should show that j T 2 ( j, j ) j = δ jj 24 / 2 2 / [(2 j ) 2 j(2 j + )(2 j + 2)(2 j + 3)]. Next we have an extremely useful result, the operator replacement theorem, which is used to replace the exact operator, for which the matrix elements are unfamiliar or tedious to evaluate, by a simpler operator. This operator replacement is only valid for restricted conditions. The operator replacement theorem: nj T ( S) n j njm j T p ( S) n j m j = njm j T p ( R) n j m j. nj T ( R) n j This implies that the matrix elements of the difficult operator, T ( S), are proportional to those of the easy operator, Tp ( R). This is especially useful when R = j, because matrix elements of any Tp p () in j

4 5.74 RWF Lecture #4 4 4 njm j the basis are diagonal in j and all nj T ()nj j are nown. This operator replacement, however, can only be used for j = 0 matrix elements of T ( S). Derivation of standard operator replacement for H SO : p SO H = ξ( r i )l i s i sum is over spin-orbitals. i This is a very inconvenient form for evaluating matrix elements of many-electron basis states. JLSM J ζ( )l i s i J L S i r i J = JLSM J ζ( r i )l i s i J L S J J L S J s i i J L S J J L S J. completeness For every value of i, the operator ξ(r i )l i satisfies the commutation rule definition of a T q vector operator with respect to L and J and the operator s i satisfies the definition of T q with respect to S and J. So we can use the operator replacement theorem twice SO JLS ξ( r i )l i J L S JLS s i JLS JLSM J H J L S J = JLS L J L S JLS S J L S i J L S J JLSM J L J L S J J L S S S J L S J. The components of the operators L and S are all diagonal in the L and S quantum numbers, which means that the operator replacement is only valid for L = S = 0 matrix elements of H SO. The product of ratios of reduced matrix elements collapses to a single constant because the cofactor of the term in [ ] is independent of i and we can carry out the completeness sum to contract the matrix element to its nearly final form: JLSM J J LS J = ζ(nls) JLSM J L S J LS H SO J where JLS ξ( r i )l i J LS JLS s i JLS ζ(nls). JLS L J LS JLSS J LS i J

5 5.74 RWF Lecture #4 4 5 However, since L S = 2[J 2 L 2 S 2 ], which is diagonal in J and M J, SO JLSM J H JLSM J = ζ(nls) JLSM J L S JLSM J = ζ(nls )[ J( J + ) L( L + ) S( S +) ]. 2 This is a reduced, but extremely convenient form of H SO. It does not tell us how to evaluate L = ±, S = ± matrix elements, but we do not need to now about those matrix elements when the energy separation between different L S terms is much larger than the spin-orbit splittings of each L S term into its J components. Now we want to use the W-E theorem to evaluate the reduced matrix elements of composite systems. The H SO case was an example of a composite system. Case (i). An operator is a sum of operators, each one operating on a different angular momentum, electron, or atom. We are interested only in matrix elements of the system operator. j = j + j 2 jjjm 2 J Tq ( A ) A operates only on system. Evaluate in the coupled representation: J M J J J jjjm 2 J T q ( A ) jjjm 2 J = ( ) jjj 2 T ( A ) jjj 2. MJ q M J Now evaluate the same matrix element by transforming to the uncoupled representation. This generates a separated subsystem reduced matrix element T A jjjm = δ j j δ m m ( ) j m 2 2 J m, m, m jjjm J q ( ) 2 j j jjmm 2 2 JM J jjmm 2 2 JM J j T ( A ) j. m q m Thus we obtain a relationship between the coupled and uncoupled basis reduced matrix elements.

6 5.74 RWF Lecture #4 4 6 j 2 J 2 ( ) jj 2 J = δ j 2 j 2 jj J T A ( ) j [(2J + )( 2J +)] j J J j T ( A ) j. J j The uncoupled basis reduced matrix elements are more fundamental, but the coupled basis reduced matrix elements are more convenient to use. This is highly relevant to your 37 Ba sd pd transition moment 2 / homewor problem. Examples: (i) expectation values of l + ()l () in a t=0 non-eigenstate pluc of a two-electron atom. (ii) transition moment (ran ) in a two-electron atom: do this. ll 2LM T L q ( µ ( )) l l 2L L = L L ( ) L M L l l L T ( µ ( )) L l l M L q M L 2 2 * We see only L L = ±,0, M L M L = q. * We get a different phenomenological transition moment for each l, l 2, L. * We do not get a l 2 selection rule. Now use reduction to single-orbital reduced matrix element * l 2 = 0, selection rule ll 2 LM LTq ( µ ( )) l l 2 L L = δ l l ( ) l l 2 ( ) L M L L L + + L + / [(2L + )( 2L +)] 2 M L q M L 2 2 l L L l T ( µ ( )) l L l single spin-orbital reduced matrix element * all L of l l transition related to single spin-orbital reduced matrix element

7 5.74 RWF Lecture #4 4 7 Silly example of 7 O p 4 ( 3 P) p 3 d Hidden Redundancy in Spectroscopy I = 5/2 ground state of O 2 is 3 P 2 p 3 d configuration has [ 4 S] 3 D, 5 D [ 2 D] S, 3 S, P, 3 P, D, 3 D, F, 3 F, G, 3 G [ 2 P] P, 3 P, D, 3 D, F, 3 F total degeneracy (excluding I = 5/2) 200 total degeneracy (including I = 5/2) values of J 8 L S multiplet states 8 different ζ(n,l,s) spin orbit constants only 2 values of ζ 2p and ζ nd p r d determines 8 transition moments (6 are non-zero) 4 magnetic hf parameters (two l I, two s I) determine hf in 38 J states The atomic orbital electronic parameters exhibit explicit n-scaling. The many electron parameters conceal this fundamental simplicity.

8 5.74 RWF Lecture #4 4 8 Case (ii) An operator is a product of two operators, each one acting on a different sub-system. We K A 2 q [ 2 are going to want to relate reduced matrix elements of the composite operator, T T ( ), T (A )], and the sub-system operators, T ( A ) and T 2 ( A2), q q 2 j j 2 J jjj T K [ / T ( ), T (A )] jjj = [(2J + )( 2 + )( 2 J +)] 2 2 A K j j 2 J ( ) T 2 2 2) 2 j T A j j (A j. The 9-j coefficient results from the transformation between the (( j, j ) J,(, 2) K) J 2 coupling scheme to the (( j, ),( j, 2 ) j )J coupling scheme. j 2 2 Example: off-diagonal matrix elements of ai S I+ S=G 0 [ I G = T T (), I T ( S ) ISG T K [T (), I T ( S )]IS G I S G 2 / = [(2G +)( 3)( 2G +)] 0 I S G IT () I I ST ( S ) S. This is a silly example because I cannot change for an atom and ST ( S ) S = ST ( S ) S δ ss. ] A much better example would be an interaction between 2 ran 2 (quadrupolar) quantities. Case (iii) An operator is a composite of two operators, both of which operate on part of a,2 coupled scheme. We need the relationship between the T (A,B ) and T K T Q ( A, B ) ) / (2K +) ( ( A ), T 2 (B ) reduced matrix elements, +Q K q q2 Q q 2 qq 2 T q (A )T2 (B ).

9 5.74 RWF Lecture #4 4 9 The relationship between reduced matrix elements, obtained by applying the Wigner-Ecart Theorem to both sides of the above operator-uncoupling equation and inserting completeness between T q ( A ) and T B ), is q 2 ( 2 K 2 K + J + J nj T ( A,B ) n J = (2K +) / ( ) 2 K T 2 nj T ( A ) nj nj (B ) n J n J J J J H SO = ξ( r i )l i s i has the form of T K (l i,s i ) where K = 0 (l s is a scalar operator with respect to j). i i Anomalous Commutation Rules and Consequences The normal commutation rule, which we use to define an angular momentum, is [J, J ] = ih ε IJK J K. I J This applies only to laboratory fixed components. It is necessary to derive the commutation rules for the molecule fixed coordinate system. Some surprises result! K The spherical tensors associated with lab fixed angular momentum components are T P=0 [ J]= J Z T P=± [ J ]= m2 2 (J / X ± ij Y ). You now all of the matrix elements of T [ ] on JM. P J However, when the commutation rules for molecule fixed angular mometum components are derived, we find [ J i, J j] = ih ε ij J.

10 5.74 RWF Lecture #4 4 0 This turns out to imply that J ± 2 / JΩM = h[j ( J + ) M( M m )] JΩ m M J ± acts as a lowering operator. This anomalous behavior applies for J, N = J S, R = J L S but not for L, S. We must be careful about how we construct molecule-fixed spherical tensor operators out of J. They do not have the expected form. To find out what form they do have, it is necessary to transform between molecule and lab-fixed systems. We get a useful redefinition of molecule fixed components of J. q = 0 Ω = Ω () ω J q ( ) P T P ( JD ) Pq ( )* P 2 q JJ JΩ MJ J Ω M ( ) J Ω J J / = [J( J+)( 2J +)] δ δ Ω q Ω mm q = +2 /2 [J(J+) Ω(Ω+)] /2 Ω Ω = q = 2 /2 [J(J+) Ω(Ω )] /2 Ω Ω = which is the naively expected form for matrix elements of raising and lowering operators. When we form a scalar product with this form of J and other normal operators, we must remember to write JP = J T [ P] rather than the usual expression for a scalar product of two T operators, q q T ( A ) T ( B ) = ( ) P T A T p ( ). p= p ( ) B