A spontaneous wavefunction collapse thought experiment

Question

Suppose $^\dagger$ we have absolutely no knowledge of a hydrogen atom. It is well isolated. And we put a photon detector next to it (and pretty sure it has no interaction with the hydrogen atom.)

Now suppose now the detector receives a photon with a specific frequency, which indicates the hydrogen atom was at a specific energy eigenstate {$n'$}, and now, {$n'-1$}.

The question is: could the hydrogen atom be in a superposition of energy eigenstates?

If yes $^{\dagger\dagger}$, then since the specific frequency indicates that the hydrogen atom now is in a specific energy eigenstate; therefore, it must have been experiencing a wavefunction collapse. And since we premised no interaction or whatsoever (there is only a equally well isolated detector next to the hydrogen atom, or rather we could place the detector $3\cdot 10^8m$ far away... ); therefore, there must be a spontaneous wavefunction collapse.

I wonder, what are the problems with my thought experiement? $^{\dagger\dagger\dagger}$

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$\dagger :$I am attempting to show that wavefunction collapse spontaneously; however, I am looking for the flaw of it, as I have just learned my undergrad quantum physics last school year, so I probably missed something here.

$\dagger\dagger$: I have asked this question to some people and place, and I was told yes. So, I will just assume yes here.

$\dagger\dagger\dagger$ : I have asked a question here, and have also read that spontaneous wavefunction collapse is nearly impossible. so I wonder what are the flaws in my thought experiment? Here are a few guesses of mine:

1.) I basically assume measurement is about interaction, however, it can be related to entanglement. I wonder if it is exactly the flaw.

2.) On the other hand. I read a article some time ago, it basically says the whole thing about emitting photon is deterministic, and has something to do with field; however, sadly, I do not quite understand the article, but I doubt it also possibly be the problems with my thought experiment.

Would anyone be kind enough to shed some light on this matter?

Answer 1

The resolution is that the wavefunction collapse isn't spontaneous at all -- it's due to the interaction of the atom with the electromagnetic field. This interaction is nonzero even though the classical field value could be zero everywhere.

The electromagnetic field is often treated classically during a first exposure to QM. Then the interaction term would be something like $$H_{\text{int}} \sim \mathbf{d} \cdot \mathbf{E}$$ where $\mathbf{d}$ is the operator for the dipole moment of the atom, and $\mathbf{E}$ is the classical electric field. Then indeed there is no interaction when $\mathbf{E} = 0$.

However, a more thorough treatment should also quantize the electromagnetic field. For simplicity, let's suppose we're in a resonant cavity with only one mode of the electromagnetic field; we can then treat its states as those of a harmonic oscillator, where $|n\rangle$ is the state with $n$ photons, with raising and lowering operators $a^\dagger$ and $a$. Similarly, for simplicity, treat the atom as a harmonic oscillator as well, with raising and lowering operators $b^\dagger$ and $b$. Then the interaction term is $$H_{\text{int}} \sim a^\dagger \otimes b + \text{h.c.}$$ That is, the atom can emit a photon, increasing the number of photons in the field, or absorb a photon, increasing its energy level.

This interaction term is always nonzero even when the initial state of the electromagnetic field is the vacuum, $|0 \rangle$. This allows for "spontaneous" emission, where the atom emits even though no photons are around. However, the name's a misnomer -- it's not really spontaneous, it's due to the interaction of the atom with the (quantum) electromagnetic field. (One way to see this is to think of $\mathbf{E}$ as a classical field, but 'smudged' about its zero value by the uncertainty principle.) If you only ever look at the atom's state, it's the emission of this photon that collapses the superposition.