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When an electron falls from an energy state to a lower one, electromagnetic radiation is emitted. Is this equally emitted in all directions (as a spherical wave) and can we only give it a direction after it is detected at some location in the form of a photon, or is it/can it be emitted in a more definite direction?

If the first case is correct, is it true that after emitting the photon the change in momentum of the electron (or the system to which it belongs) is a spherically symmetric superposition which collapses into a definite direction when the photon is received somewhere?

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This is related to Photons from stars--how do they fill in such large angular distances? and also to Some doubts about photons. The answer is that as long as the experiment is spherically symmetric, e.g. you don't have external influences like an electric field, your summary is correct.

I'm not sure how useful it is to describe the light as a photon, because the photon description is most useful to describe how light interacts rather than how it propagates. However if you're determined to describe the light as a photon then it would initially be in a superposition of all momentum directions, and so would the atom it is emitted from. A particular value for the momentum would emerge only when you collapsed the superposition by making a measurement.

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Is this equally emitted in all directions (as a spherical wave) and can we only give it a direction after it is detected at some location in the form of a photon, or is it/can it be emitted in a more definite direction?

Depends on a theory. In semi-classical radiation theory, neo-classical theory or in stochastic electrodynamics, radiation is described by continuous EM field. The radiation from one transition should be approximately that of a dipole - it has polarization and angular characteristics of a dipole radiation. The phase of the radiation moves out as spherical wave, but the intensity is strongest in the plane perpendicular to the transition dipole matrix element $\boldsymbol{\mu}_{12}$.

In theory with photons, my understanding is that mathematics is similar to the above but the field has only probabilistic meaning. When the photon is absorbed at one place, the probabilistic field all around the molecule is discarded. Which brings lot of infamous problems to common sense, like how detection here can affect what happens there etc.

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