04_lecnotes_rwf

04_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 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 | ΑΜ Α and | ΒΜ Β angular momentum basis states to form coupled | ΑΒ C Μ C basis states.
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2 3 5.74 RWF Lecture #4 4 – 2 j m j j k j ′  k k njm j q ( ) T A j ( nj T A ( ) j n ′ ′ m j n ′ ′ = − 1) m j q m j The 3-j coefficient is what you would expect for coupling j 1 = j , m j 1 = m j j 2 = k m j = q , to 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 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): 1 T j 0 ( ) = j z 0 2 T j j 0 ( , ) = 2 1 2 2 2 ± T j j 0 ( , ) = 6 / ( 3 j j ) . z Using the W-E theorem j m j 1 j ′  1 1 jm j T j 0 ( ) m ( j T j ( ) j , j j = − 1) m j 0 m j but we also know jm j z j m j j jj = δ δ m j m j m j . Thus, using the analytical expression for the 3-j coefficient 1 2 m j 0 m j ( ) j m j m j ( j + 1 2 j + 1) ] , j 1 j = − 1 j [ )( / we evaluate the reduced matrix element: 1 )( 1 2 / j T j ( ) j ′ = δ jj [ j ( j + 1 2 j + 1) ] .
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