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Suppose that an electron with spin up emits a photon in the field of an ion (bremsstrahlung). What is the spin of the emitted photon? Is it correct to say that the photon is circularly polarized if the spin of the electron flips down and linearly polarised if it remains up?

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Total angular momentum is always conserved. If there's nothing else changing (like the orbital angular momentum of the electron around the ion), then what you said sounds correct, neglecting details like the direction of propagation of the light vs. the axis of quantization of the spin. But in general, there WILL be other stuff going on as well (such as the aforementioned orbital angular momentum, etc. etc.).

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Spin of the photon is always 1 as it is a spin 1 particle. If you are asking about projection of spin into some direction in the center of mass reference frame then this has no meaning (because photon is massless). Instead, one needs to project into the direction of motion (by forming a scalar product of the spin operator with momentum) and this is called helicity.

Now, by conservation of angular momentum it must always be the case that the electron's spin flips in the first order (because $1/2 = -1/2 + 1$ is the only possible solution) and that means the photon is circularly polarized. In the second order, it could stay the same if you emitted two photons with opposite helicities one after the other.

So, you can actually measure the projection of the spin of the electron by measuring the polarization of the photon (which should be circular). If this experiment is however done over the sample which has electrons in both states, then one needs to average over that and you obtain (circularly) unpolarized photons.

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