There are a lot of questions on this site about annihilation and quantum processes (and whether they are instantaneous), but none of them specifically give a satisfactory answer to my question. None of these specifically take annihilation to be instantaneous or not, and none of them specifically address how to model the process smoothly.

Like this one:

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> We then talk about a tunneling event or an emission/absorption event, and we can attribute to them a characteristic time.
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[Are quantum events such as decay, emission, absorption and tunneling truly instantaneous or is there some small time period?](https://physics.stackexchange.com/questions/690586/are-quantum-events-such-as-decay-emission-absorption-and-tunneling-truly-insta)

I have read this question:

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> But there is no clearly identifiable "particles" in the middle when the scattering reaction takes place, just a messy and evolving quantum state. We can track neither experimentally nor theoretically the identity of a particle in the input and point to any specific point in time when it "disappears". QFT allows us to compute the probabilities for certain inputs to lead to certain outputs, but it is not tractable to produce some sort of "live view" of the scattering process - it's like a black box, particles go in, particles come out.
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[At what point of approach in annihilation process the charges of antiparticles disappear? How is relevant to the electron-proton interaction?](https://physics.stackexchange.com/questions/701762/at-what-point-of-approach-in-annihilation-process-the-charges-of-antiparticles-d)

Now I am not asking about the intermediate states or what is inbetween. I am simply asking, if nothing is inbetween, just a "black box", and electron-positron pair coming in, and two photons (or a photon near a nucleus, but let's not complicate it with this) going out, then is this process truly instantaneous?

The reason I am asking, is because as far as I understand, quantum mechanical processes always take time and are never instantaneous. The deeper reasoning behind this should have to do with the speed of light not being infinite, but my question is not about this. My favorite example of describing whether quantum processes are instantenous is my previous question about the electron's change of energy levels (in atoms).

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> In every reasonable interpretation of this question, the answer is **no**. But there are historical and sociological reasons why a lot of people say the answer is yes.
> Consider an electron in a hydrogen atom which falls from the $2p$ state to the $1s$ state. The quantum state of the electron over time will be (assuming one can just trace out the environment without issue)
> $$|\psi(t) \rangle = c\_1(t) |2p \rangle + c\_2(t) | 1s \rangle.$$
> Over time, $c\_1(t)$ smoothly decreases from one to zero, while $c\_2(t)$ smoothly increases from zero to one. So everything happens continuously, and there are no jumps. (Meanwhile, the expected number of photons in the electromagnetic field also smoothly increases from zero to one, via continuous superpositions of zero-photon and one-photon states.)
> The wavefunction $\psi(\mathbf{r}, t)$ evolves perfectly continuously in time during this process, and there is no point when one can say a jump has "instantly" occurred.
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[Do electrons really perform instantaneous quantum leaps?](https://physics.stackexchange.com/questions/488527/do-electrons-really-perform-instantaneous-quantum-leaps)

Now this in knzhou's perfect answer is really naively understandable, the quantum state smoothly changes with the expectation values of the first state and the second state changing smoothly. This is understandable, because we are talking about a probability distribution of the electron around the nucleus.

However, when I try to use this for the process of annihilation, it does not seem to work. The good thing about the electron jump example is that the probability distribution describes the electron's position around the nucleus with co-existing multiple states already at certain probabilities. From there, applying the smooth change is understandable.

Though, this is where it becomes hard to apply for the annihilation, because the particle's type itself is not like that. The particle's type itself is in a definite state (as far as I understand), and does not evolve (without interaction), does not change, and an electron is always an electron. I could not say that the electron's type is described by the wavefunction in a way where the particle is at certain probabilities co-existing as an electron and as a photon at the same time.

But with annihilation, something like that should happen. The quantum state before annihilation describes the electron-positron pair's state as a definite, the type of particles are just that (before annihilation, the electron is 100% electron, and the positron is 100% positron, and after annihilation, the photons are a 100% photons). After the annihilation, though, the particle's type changes instantaneously, to two photons. There is no smooth change, no probabilities.

In other words, can we somehow smoothly model the annihilation process with the similar method (electronic transition) in knzhou's answer, or is it just an abrupt (instantaneous) change?

Question:

1. Is annihilation a smooth process that requires time?