spherical 0.1.1

Evaluate and transform D matrices, 3-j symbols, and (scalar or spin-weighted) spherical harmonics

Spherical Functions

Python/numba package for evaluating and transforming Wigner's 𝔇 matrices, Wigner's 3-j symbols, and spin-weighted (and scalar) spherical harmonics. These functions are evaluated directly in terms of quaternions, as well as in the more standard forms of spherical coordinates and Euler angles.¹

The conventions for this package are described in detail on this page.

Installation

Because this package is pure python code, installation is very simple. In particular, with a reasonably modern installation, you can just run a command like

conda install -c conda-forge spherical

or

python -m pip install spherical

Either of these will download and install the package.

Usage

First, we show a very simple example of usage with Euler angles, though it breaks my heart to do so:¹

>>> import spherical as sf
>>> alpha, beta, gamma = 0.1, 0.2, 0.3
>>> ell,mp,m = 3,2,1
>>> sf.Wigner_D_element(alpha, beta, gamma, ell, mp, m)

Of course, it's always better to use unit quaternions to describe rotations:

>>> import numpy as np
>>> import quaternionic
>>> R = quaternionic.array(1,2,3,4).normalized
>>> ell,mp,m = 3,2,1
>>> sf.Wigner_D_element(R, ell, mp, m)

If you need to calculate values of the 𝔇^(ℓ) matrix elements for many values of (ℓ, m', m), it is more efficient to do so all at once. The following calculates all modes for ℓ from 2 to 8 (inclusive):

>>> indices = np.array([[ell,mp,m] for ell in range(2,9)
... for mp in range(-ell, ell+1) for m in range(-ell, ell+1)])
>>> sf.Wigner_D_element(R, indices)

Finally, if you really need to put the pedal to the metal, and are willing to guarantee that the input arguments are correct, you can use a special hidden form of the function:

>>> sf._Wigner_D_element(R.a, R.b, indices, elements)

Here, R.a and R.b are the two complex parts of the quaternion defined on this page (though the user need not care about that). The indices variable is assumed to be a two-dimensional array of integers, where the second dimension has size three, representing the (ℓ, m', m) indices. This avoids certain somewhat slower pure-python operations involving argument checking, reshaping, etc. The elements variable must be a one-dimensional array of complex numbers (can be uninitialized), which will be replaced with the corresponding values on return. Again, however, there is no input dimension checking here, so if you give bad inputs, behavior could range from silently wrong to exceptions to segmentation faults. Caveat emptor.

Acknowledgments

I very much appreciate Barry Wardell's help in sorting out the relationships between my conventions and those of other people and software packages (especially Mathematica's crazy conventions).

This code is, of course, hosted on github. Because it is an open-source project, the hosting is free, and all the wonderful features of github are available, including free wiki space and web page hosting, pull requests, a nice interface to the git logs, etc.

The work of creating this code was supported in part by the Sherman Fairchild Foundation and by NSF Grants No. PHY-1306125 and AST-1333129.

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¹ Euler angles are awful

Euler angles are pretty much the worst things ever and it makes me feel bad even supporting them. Quaternions are faster, more accurate, basically free of singularities, more intuitive, and generally easier to understand. You can work entirely without Euler angles (I certainly do). You absolutely never need them. But if you're so old fashioned that you really can't give them up, they are fully supported.