- #1
Like Tony Stark
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- Homework Statement
- Rotate the spherical harmonic $$\ket{l=2, m=1}=Y_{2, 1}$$ an angle of π/4 about the y-axis.
- Relevant Equations
- $$\sum_{m'=-l}^{l} {d^{(l)}}_{m, m'} Y_{l, m'}$$
I tried using the Wigner matrices:
$$\sum_{m'=-2}^{2} {d^{(2)}}_{1m'} Y_{2; m'}={d^{(2)}}_{1 -2} Y_{2; -2} + {d^{(2)}}_{1 -1} Y_{2; -1} + ...= -\frac{1-\cos(\beta)}{2} \sin(\beta) \sqrt{\frac{15}{32 \pi}} \sin^2(\theta) e^{-i \phi} + ...$$
where $$\beta=\frac{\pi}{4}$$. But I don't know if this is ok since $$\beta$$ is an Euler angle while $$\theta$$ and $$\phi$$ are not. If this is not right, what should I do?
$$\sum_{m'=-2}^{2} {d^{(2)}}_{1m'} Y_{2; m'}={d^{(2)}}_{1 -2} Y_{2; -2} + {d^{(2)}}_{1 -1} Y_{2; -1} + ...= -\frac{1-\cos(\beta)}{2} \sin(\beta) \sqrt{\frac{15}{32 \pi}} \sin^2(\theta) e^{-i \phi} + ...$$
where $$\beta=\frac{\pi}{4}$$. But I don't know if this is ok since $$\beta$$ is an Euler angle while $$\theta$$ and $$\phi$$ are not. If this is not right, what should I do?
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