Consider an angular momentum eigenstate $|j,m \rangle$, what would be the simplest way to model a rotation of this state in Python which starts out in a state where $m = j$ and is acted on by a rotation operator which rotates about the $y$ axis, hence $e^{i\frac{\pi}{4}\hat{J}_{y}}$.

My idea it to express $|j,m\rangle$ as a column vector $\begin{pmatrix}1 \\ 0 \\ 0\\ \cdot \\ \cdot \end{pmatrix}$ and then try to express $e^{i\frac{\pi}{4}\hat{J}_{y}}$ as a matrix? Hence $e^{i\frac{\pi}{4}\hat{J}_{y}}\begin{pmatrix}1 \\ 0 \\ 0\\ \cdot \\ \cdot \end{pmatrix} = \begin{pmatrix}\cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot\\ \cdot \\ \cdot \end{pmatrix} $ $\begin{pmatrix}1 \\ 0 \\ 0\\ \cdot \\ \cdot \end{pmatrix}$ would given the desired column matrix. What do you think? Do you maybe know how to code this matrix?


closed as off-topic by AccidentalFourierTransform, ACuriousMind Jun 3 '17 at 13:56

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    $\begingroup$ I'm voting to close this question as off-topic because it's about writing code and not about physics per se. $\endgroup$ – AccidentalFourierTransform Jun 3 '17 at 13:23
  • $\begingroup$ @AccidentalFourierTransform Just leave it for a while!!! It's under 'computational physics' tag. $\endgroup$ – user100411 Jun 3 '17 at 13:24
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    $\begingroup$ computational-physics is a tag for asking about algorithms and numerical methods, not for requesting specific code. It's also a bit unclear what exactly your issue is - if you just want to know what matrix corresponds to $\mathrm{e}^{\mathrm{i}J_y}$, then that's not a question about computational methods at all, it's just a QM/linear algebra question. $\endgroup$ – ACuriousMind Jun 3 '17 at 13:56
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    $\begingroup$ Moreover these matrices have known analytic form in terms of Jacobi polynomials: see en.m.wikipedia.org/wiki/Wigner_D-matrix. $\endgroup$ – ZeroTheHero Jun 4 '17 at 14:18
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    $\begingroup$ Right... for small $j$ this will work but the computational overhead grows rapidly as $j$ increases. If you start with $m=j$ you get an angular momentum coherent state and there are simplified expressions based on the normal ordering of the lowering and raising operators that come handy. There are also useful recursion relations. See Wolters, G. F. "Simple method for the explicit calculation of d-functions." Nuclear Physics B 18.2 (1970): 625-653. $\endgroup$ – ZeroTheHero Jun 5 '17 at 1:11