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The easiest way to see that $p^4$ is spherically symmetric is to view it in momentum space. If you apply $p^4$ to a momentum eigenstate $|p\rangle$ the result clearly only depends on the magnitude of the momentum vector of the state and not on its orientation, so if $R$ is a rotation operator we have p^4 R|p\rangle = Rp^4|p\rangle ...

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Let us see how the notion of the "electron cloud" enters the calculations. In the first Born approximation, the elastic scattering is determined with the atomic form-factor $F$ containing the reduced mass in the wave function and the "electron coordinate" in the exponential: $$d\sigma\propto|Zf(\mathbf{q})-F(\mathbf{q})|^2,\qquad (1)$$ F(\mathbf{q})=\int ...

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Probably too late now, but I have to answer this. There is indeed a way to predict the direction of a scattered Lyman $\alpha$ photon. The answer depends on whether the scattering takes place in the core or the wings of the line. In the core (i.e. closer to the line center than about 3 Doppler widths), we can use the dipole approximation, so the phase ...

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