[duality]

"An atom can be described by its quantum state only if it's isolated and in that case its energy is constant."

How do you figure? As a contradiction, take your atom+electromagnetic field system, describe the transition from excited atom to unexcited atom + photon state, and project out the E&M field. Voila, now you have a quantum description of an atom transitioning between different energy states. Its dynamics may look funny, i.e. they may appear nonlocal, they may not conserve energy, etc. but that's different from saying "there is not a quantum description of these dynamics" which is what you're claiming.

[kgwgk]

https://en.m.wikipedia.org/wiki/Quantum_state

> project out the E&M field. Voila, now you have a quantum description

There is no wavefunction describing the state of the subsystem because the system is not separable.

> Its dynamics may look funny, i.e. they may appear nonlocal, they may not conserve energy, etc. but that's different from saying "there is not a quantum description of these dynamics" which is what you're claiming.

What is the “quantum description of these dynamics”?

Quantum mechanics is usually based on something like the following postulate: “The state of a physical system is described by a well-behaved function of the coordinates and time, Ψ(q, t). The function contains all the information that can be known about the system.”

[gaze]

The system is separable after you measure out your ancilla. Imagine a stream of qubits B_1, B_2, ... prepared in +X who interact weakly with qubit A prepared in +X. and are then strongly measured projectively. After the brief interaction, A and a certain B will be entangled. Projectively measuring this B updates our knowledge of A, but due to the weak entanglement, the adjustment is small and incremental. The state of such B qubit is known and thus there is no remaining entanglement between A and B, and thus A and B are separable. After many such Bs, an outside observer privy to all measurement outcomes will understand that A is doing an absorbing random walk to the poles of the Bloch sphere. Throwing away this information and taking an ensemble average, you'd see an exponential decay of coherence.

This is a toy model of a qubit being weakly measured by an incident flying field.

[kgwgk]

> Throwing away this information and taking an ensemble average, you'd see an exponential decay of coherence.

That is described by a density matrix but is not a pure state. The true state may be a pure state (because as you said the systems are separable after the measurement) but our description is a mixture reflecting our imperfect information (and not a superposition of the pure states corresponding to the potential outcomes).

[duality]

Not sure exactly what you're linking the Wikipedia article on quantum states, but if you like it as a reference, check out https://en.wikipedia.org/wiki/Quantum_state#Mixed_states. The operation of "projecting out" the E&M field in my previous comment would be realized on the density matrix as a partial trace over the electromagnetic field. You can also take the partial trace of this field of the Hamiltonian operator to see the effective dynamics of the atom when "ignoring" the E&M field. This is a fully quantum description of the state, so I stand by the statement that your claim "An atom can be described by its quantum state only if it's isolated and in that case its energy is constant." is incorrect.

[kgwgk]

Do you agree that “The state of a physical system is described by a well-behaved function of the coordinates and time, Ψ(q, t). The function contains all the information that can be known about the system.”?

When the atom is coupled with the electromagnetic field and the state of the system is not separable there is no complete description of the atom given by a wavefunction defining its quantum state. You can have an incomplete description by tracing out the rest of the system, I agree.

Let’s say then that "An atom can be completely described by its quantum state only if it's isolated and in that case its energy is constant."

Edit: in any case, my point was (and I think that we will agree) that it is misleading to say “Consider a system that transitions from energy state 0 to an adjacent energy state 1. [...] To go smoothly from 0 to 1, the system transitions through a series of superpositions of both states”.

The atom goes from the state 0 to the state 1 but during the transition it’s not described by a superposition of those states (that would be a pure state). If anything, it is described by an (improper) mixture of those states, obtained by tracing out the rest of the system.

[lisper]

Yes, but note that I deliberately used the word "system" rather than "atom". The system is the combination of the atom plus whatever it's absorbing energy from or emitting energy to. And that (entangled) system is in a superposition.

[kgwgk]

Ok, I was confused because if you say “Consider a system that transitions from energy state 0 to an adjacent energy state 1” it sounds as if the energy of the “system” is changing and when you say that “a particle [...] can be in two different energy states at the same time” it seems that you are referring to the atom being in a superposition of states with different energy.

[lisper]

One can speak meaningfully of "an atom in a superposition of energy states" despite the fact that, strictly speaking, such a thing is not possible, just as one can speak meaningfully of "the force of gravity" despite the fact that, strictly speaking, there is no such force. The latter is understood as the force-like effect of curved spacetime, and the former is understood as "an atom being a component of a system in a superposition of states with different distributions of energy" (or something like that). Communications between humans becomes more productive when we cut each other a little terminological slack.