MIT Department of Chemistry 5 74 Spring 2004 Introductory Quantum Mechanics II Instructor Prof Robert Field 5 74 RWF Lecture 4 4 1 The Wigner Eckart 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 Eckart 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 Eckart 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 Eckart 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 Eckart Theorem may be viewed as a generalization of the coupling of separate and angular momentum basis states to form coupled C C basis states 5 74 RWF Lecture 4 4 2 k njm j Tq A n j m j 1 j m j j m j k j k nj T A n j q m j The 3 j coefficient is what you would expect for coupling j1 j m j1 m j to j2 k m j 2 q to form j3 j m j 3 m j The factor with the two pairs of vertical lines is called a reduced matrix element It 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 mj j basis state 1 T0 j jz 0 2 2 1 2 T0 j j j T0 j j 6 3j 2 z 2 j Using the W E theorem 1 jm j T0 j j m j 1 j m j m j but we also know jm j jz j m j jj m j m j m j Thus using the analytical expression for the 3 j coefficient j m j 1 j j m j 1 2 1 m j j j 1 2 j 1 0 m j we evaluate the reduced matrix element j T j j jj j j 1 2 j 1 1 Another example 1 2 1 j 1 j T j j 0 m j 5 74 RWF Lecture 4 4 3 0 jm j T0 j j j m j 1 1 j j m j m j j m j 0 j T j j 0 m j 0 j jj m j m j m j j 0 j T j j 0 m j 0 0 j 1 j m 2 j 1 1 2 0 mj But m j thus jm j T00 j j j m j jj m j m j 2 j 1 1 2 j T 0 j j j But when the matrix element is evaluated directly we have 2 jm j j j m j jj m j m j j j 1 thus 0 j T0 j j j 2 j 1 1 2 j j 1 You should show that j T 2 j j j jj 24 1 2 2 j 1 2 j 2 j 1 2 j 2 2 j 3 1 2 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 k k njm j Tp S n j m j nj T S n j k nj T R n j k njm j Tp R n j m j k This implies that the matrix elements of the difficult operator Tp S are proportional to those of the k k easy operator Tp R This is especially useful when R j because matrix elements of any Tp j in 5 74 RWF Lecture 4 4 4 k the njm j basis are diagonal in j and all nj T j nj are known This operator replacement however can only be used for j 0 matrix elements of Tpk S Derivation of standard operator replacement for HSO H SO ri l i si sum is over spin orbitals i This is a very inconvenient form for evaluating matrix elements of many electron basis states JLSM J ri l i si J L S J i i J L S J JLSM J ri l i si J L S J J L S J si J L S J completeness For every value of i the operator ri li satisfies the commutation rule definition of a Tq1 vector operator with respect to L and J and the operator si satisfies the definition of Tq1 with respect to S and J So we can use the operator replacement theorem twice JLSM J H SO J L S J i J L S J JLS ri l i J L S JLS L J L S J L S si J L S J L S S J L S 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 HSO 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 HSO J LS J nLS JLSM J L S J LS J where NLS i J JLS ri l i J LS J LS si J LS JLS L J LS J LS S J LS 5 74 RWF Lecture 4 4 5 1 However since L S 2 J2 L2 S2 which is diagonal in J and MJ JLSM J H SO JLSM J nLS JLSM J L S JLSM J 1 nLS J J 1 L L 1 S S 1 2 This is a reduced but extremely convenient form of HSO It does not tell us …