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I have a newbie question that I am trying to wrap my brain around.

Single photon gamma emission from a nucleus undergoing a $2^{+}$ to $0^{+}$ transition would involve an emitted photon with angular momentum of $2 \hbar$ in order for angular momentum to be conserved.

However, photons are spin one particles, so the above photon must have some orbital angular momentum in addition to its spin angular momentum.

I have difficulty visualizing how an unbounded, massless particle could have orbital angular momentum. Unless of course it were orbiting a black hole.

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    $\begingroup$ Interesting. So the wavefunctions of photons with orbital angular momentum aren't simply just plane waves. The phase of the wavefunctions will have a sinusoidal dependence on the polar angle (in cylindrical coodinates) similar to the helical modes of light beams with orbital angular momentum. Integrating over these wavefunctions then gives a net angular momentum $l_{z}$ for the chosen direction. Therefore not dependent on the origin indicated for one's coordinate system. $\endgroup$ – dualredlaugh Jul 20 '16 at 3:32
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I commented on the question but on advise, I am promoting this to an answer. A particle moving in a straight line can have non zero orbital angular momentum if the origin of the coordinate system doesn't lie on that straight line. There can also be helical modes through which light can have non zero orbital angular momentum. The first one is origin dependent, the second one isn't. And yes, in the second case, these aren't plane waves.

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Visualizations are not easy when one is dealing in the relativistic regime. One has to do the mathematics. In this case define angular momentum within special relativity. This has been done but needs to read through the maths.

Intuition for special relativity rests on the equivalence of mass/energy. The photon has zero mass and angular momentum can have no description in classical mechanics terms. Special relativity has to be used and the angular momentum is with respect to the energy/momentum carried away by the photon.

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