re: Rotations`

• To: mathgroup at yoda.physics.unc.edu
• Subject: re: Rotations`
• From: wmm at chem.wayne.edu (Martin McClain)
• Date: Thu, 23 Dec 93 13:53:09 EST

```Dear MathGroupers-
The spherical harmonics, the Wigner rotation matrix, and the  3J,
6J, and 9J coupling coefficients possess a very strictly defined
convention, the Condon and Shortley convention, which has now been adopted
by nearly everybody, including Mma.  The Wigner matrix is a function of the
Euler angles, which are therefore a part of the Condon and Shortley
convention.  To be internally consistent with its own angular momentum
quantities Mma should use this convention for its Euler matrix
RotationMatrix3D, defined in the package context Geometry`Rotations`.
However, this package appears to be based on Wigner's rotation convention,
which is anathema.  Wigner's book was a great book, but he uses a left hand
rule to define the positive sense of rotations, among other things.  This
is at the root of recent complaints about the Rotations package.

The Euler-Wigner connection should be establishable through the
famous fundamental equation for the rotation of spherical harmonics:

Y[L,m,EulerOp[alf,bet,gam][theta,phi]] ==
Sum[Wigner[L,m,k,alf,bet,gam]*Y[L,k,theta,phi],{k,-L,+L}]

where {alf,bet,gam} are the Euler angles. This equation is the basis of all
spherical tensor developments. Details are given in Edmonds, Angular
Momentum in Quantum Mechanics, a very reliable book that follows the Condon
and Shortley convention.  It is an unnumbered equation at the bottom of
page 56, where his u[j,m] is a generalization of Y[L,m].

I have put in some time trying to establish this connction myself,
but so far the numerical value of the left side is not the same as the
right side.  The problem may be with EulerOp, which involves the Euler
matrix, but is is not identical to it, and is not too definitely defined.
Or it could be a mistake I have made.  I doubt that there is any mistake by
Edmonds.  I would be glad to share my efforts in Notebook form with anybody
who would like to work on this.

Regards-  Martin McClain, Wayne State Univ, Detroit

```

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