Rotation conventions

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Rotation conventions are a veritable minefield. It is truly remarkable how much confusion the subject has amassed over the years. Spinach uses the conventions specified by Brink and Satchler in their book on angular momentum. The book is included with the kernel (see kernel/conventions). The relevant pages are reproduced below.

Sense and sign

Spinach uses the active convention - it always rotates the spin system, not the coordinate axes. Positive rotation is counter-clockwise, as shown in the figure below.

An active rotation by a positive angle. Reproduced from Brink and Satchler.

The function that builds rotation matrices is euler2dcm.m; it uses the Euler angle convention described below.

Euler angles and Wigner functions

Spinach uses active ZYZ convention described in Page 20 of Brink and Satchler (reproduced below). Note particularly the text immediately after Equation 2.12 - the rotation specification in which the axes do not drift is easy to visualize.

ZYZ convention for Euler angles and the associated definition for Wigner D matrices. Reproduced from Brink and Satchler.

Equation 2.13 is used by wigner.m to build Wigner D matrices.

Version 2.2, authors: Ilya Kuprov