Energy levels in simple finite systems are indeed quantized, but this does not mean we can not make the energy quanta be continuously parameterized. For instance, if your system is two mirrors facing each other and you are using the quantum description of the light trapped between these mirrors, you can pick any real value for the energy separation between levels of this system simply by continuously varying the distance between the mirrors.
Maybe one can make the argument that position itself is quantized (thus the position of the mirrors can not be varied continuously), but we do not have experimental reasons to believe space is discrete (and quantum mechanics does not require it to be discrete). And while it is pleasing to imagine it discrete (it is more "mathematically elegant"), we do not have any significant rigorous reasons to believe it is.
Edit: Moreover, if you want to describe (in quantum mechanics) the interaction between a finite system and the open environment around it, the only way to get a mathematical description that matches real-world experiments is to have continously parameterized energy levels for the systems making up the open environment. If you assume that only discrete values are possible, you will simply get the wrong result. Most quantum optics textbooks have reasonably good discussion of this. E.g.:
Quantum Optics by Walls and Milburn
Quantum Optics by Scully and Zubairy
Methods in Theoretical Quantum Optics by Barnett and Radmore
> this does not mean we can not make the energy quanta be continuously parameterized
Sure, but can you measure those continuously-parameterized energies? I don't see how.
Continuously parameterized energies are no different from continuously parameterized space. They are part of the mathematical model we use to make accurate predictions, but we have no direct access to either, and (AFAICT) we cannot possibly have access to them because that would violate the no-cloning theorem.
I am not sure I follow. Here is my attempt to respond to what you raised but feel free to redirect me if I misunderstood.
The following is a (simplified, abstracted) way to measure an arbitrary energy value.
1. Set the system up so that the carrier of the energy is a photon (e.g. let the two-level system decay[1] or use some form of transduction or whatever).
2. Send that photon to pass by two semi-transparent mirrors at a certain (continuously parameterized) distance between each other.
3. If the photon passes through both mirrors (as detected by a photon detector at the other side), it means its energy is equal to some known constant divided by the distance between the mirrors. If it does not pass it means it has a different energy.
4. Repeat the experiment many times as you slowly vary the distance between mirrors.
I guess in point 4 there is an issue that you need to repeat the experiment with a new realization of your photon each time. Does that have bearing on the initial point being discussed?
You are probably seeing this at this point, but just for completeness: this technique is no different from tuning a musical instrument with a tuning fork.
I papered over some details about whether we want to detect transmission or reflection and exactly what type of transparency the mirrors need to be, etc.
[1] Funnily, the usual way in which someone would prove that decay can happen at all does rely on the existence of a continuous spectrum of energies. This is the same topic I raised above when citing Quantum Optics textbooks.
> this technique is no different from tuning a musical instrument with a tuning fork.
Yes, I get that. But there are two problems. First, you cannot tune an instrument precisely. Precise tuning of a real musical instrument isn't even a meaningful concept because any wave with finite temporal extent has non-zero bandwidth. It's the same with energy. The exact same uncertainty relationship between frequency and time produces the Heisenberg uncertainty relation between energy and time, so it is not possible to produce an isolated photon at a known time with a known energy. The best you can do is produce a lot of photons so you don't have to wait forever for one of them to arrive at your detector. So the problem with the setup you describe is that in step 2 the concept of "that photon" is not well defined.
Second...
> I guess in point 4 there is an issue that you need to repeat the experiment with a new realization of your photon each time.
Yes, that too. But you need to do more than that: in order to get a meaningful result you'd need to produce photons with the same energy, i.e. you'd need to use a laser or some other kind of tuned cavity. But it is not meaningful to identify individual photons emitted from a laser because they are identical bosons.
Maybe one can make the argument that position itself is quantized (thus the position of the mirrors can not be varied continuously), but we do not have experimental reasons to believe space is discrete (and quantum mechanics does not require it to be discrete). And while it is pleasing to imagine it discrete (it is more "mathematically elegant"), we do not have any significant rigorous reasons to believe it is.
Edit: Moreover, if you want to describe (in quantum mechanics) the interaction between a finite system and the open environment around it, the only way to get a mathematical description that matches real-world experiments is to have continously parameterized energy levels for the systems making up the open environment. If you assume that only discrete values are possible, you will simply get the wrong result. Most quantum optics textbooks have reasonably good discussion of this. E.g.: