If energy is quantized, does that mean that there is a largest-possible wavelength?

Question

Given Planck's energy-frequency relation $E=hf$, since energy is quantized, presumably there exists some quantum of energy that is the smallest possible. Is there truly such a universally-minimum quantum of $E$, and does that also mean that there is a minimum-possible frequency (and thus, a maximum-possible wavelength)?

Answer 1

since energy is quantized

You have a misunderstanding here on what quantization means. At present in our theoretical models of particle interactions all the variables are continuous, both space-time and energy momentum. This means they can take any value from the field of real numbers. It is the specific solution of quantum mechanical equations, with given boundary conditions that generates quantization of energy.

The same is true for classical differential equations, as far as frequencies go. Sound frequency can take any value, and its quantization in specific modes depends on the specific problem and its boundary conditions.

There exist limits given by the value of the constants that are used in elementary particle quantum mechanical equations. It is the Planck length and the Planck time

the reciprocal of the Planck time can be interpreted as an upper bound on the frequency of a wave. This follows from the interpretation of the Planck length as a minimal length, and hence a lower bound on the wavelength.

which are at the limits of what we can see in experiments and study in astrophysical observations, but these are another story.

Answer 2

You have to keep in mind that the relation

$$ E = hf $$

holds only true for photons. Photons - generally - can have arbitrary energies, so they can have arbitrary frequencies as well.

When you think of quantization, you think of quantization of an observable for a specific system. An one-dimensional harmonic oscillator for example has the quantized energies

$$ E_\textrm{Osz} = h f_\textrm{Osz} \left(\frac{1}{2} +n\right) $$

where $n$ is $\mathbb Z_{0}^{+}$, and $f_{Osz}$ is the frequency of the oscillator! So the energy has discrete values it can hold, but remember, we are talking about the energy of the oscillator, not of a photon. If you now ask the question: What is the minimum energy (and therefore maximum wavelength) for a photon to get absorbed by the oscillator, the answer would be:

$$ E_\textrm{PhMin} = 1\cdot h f_\textrm{Osz} \ \rightarrow \ \lambda_\textrm{PhMax} = c/f_\textrm{Osz} $$

because that is the difference between two niveaus of the oscillator.

If you look at another system your answer will be different. How observables (like energy) are quantized is dependent on the system.

tl'dr:

There is no maximum wavelength for photons.

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Edit: At least not because of the Planck Relation. If there is a maximum wavelength for photons, the reasons for it will have nothing to do with the implications of your question. I could be that - at sufficiently small energies - photons show non-trivial effects which restricts another loss of energy.

(Please note that I used simplified assumptions, for instance we are in vacuum etc.)

Answer 3

Of course Anna V is correct where she says

"There exist limits given by the value of the constants that are used in elementary particle quantum mechanical equations. It is the Planck length and the Planck time the reciprocal of the Planck time can be interpreted as an upper bound on the frequency of a wave. This follows from the interpretation of the Planck length as a minimal length, and hence a lower bound on the wavelength."

And the most important is:

"which are at the limits of what we can see in experiments and study in astrophysical observations, but these are another story."

I would like to add a little explanation to these answers because I believe it can be explained in a simple understandable way. You are asking "there is a largest-possible wavelength?", which can be interpreted as "is there a smallest-possible frequency?"

1. to understand this, you have to understand where quantization comes from, how we first measured it, and how the first experiments showed to Planck, that there must be a connection between energy and frequency for photons. It all comes from the Planck-constant. This was the connection between the starting voltage (in the original experiments this was the voltage that started to excite the light emitter lamp or nowadays led), and the emitted photons frequency. Planck discovered that the different color lights that had different wavelengths, were all excited with different starting-voltages. These voltages were showing a correlation between the voltage and the wavelength. However he changed the voltage, and the color of the emitted light, the correlation was always the same, showing a constant correlation, and that (expressed in terms of photon energy and frequency) became known as the Planck-constant. It is so important, because the voltage had to do with the elementary charge, and the frequency with the energy of the photon. So if the photon emitted had only certain wavelengths, and the starting voltage was always the same constant for the same color light, then this constant could also be expressed as E=h*f, where the energy of the photon is the constant times the frequency. Why is this so important? Because the electron itself is emitting the photons, and if the emitted photons are only of certain wavelength, then it has to do with the electron's some quantity. This later became known as the electron's energy level, and since the emitted photon's where always only at some discrete wavelengths, the electron's energy level had to be discrete as well (in truth it is the difference between the different electron energy levels that corresponds to the emitted photon's energy) . This later gave rise to QM, since they wanted to somehow explain the electron's ability to stay on a certain stable quantized energy orbit level.

2. Now you see the emitting electron's energy level (in truth the difference between the different electron orbit energy levels) gives you the emitted photon's frequency if you know the Planck constant. Whys is this important, because as you ask "If energy is quantized, does that mean that there is a largest-possible wavelength?", that means that you would think that there can only be certain emitted frequency EM waves, since we only know of certain finite number of atoms, and electron levels that we have really seen experimentally. So you could think that there can only be photon frequencies, that are emitted by electrons that we know and have seen in the experiments. That would of course give you the answer yes, there is a maximal wavelength.

3. The problem is, in the universe there might be so much more then what we have seen in the experiments. Theoretically, there can be an electron energy level, that is even higher, and higher and so on, into infinity. And there you would get theoretically that your emitted photon would have 0 wavelength (so infinite frequency, so bigger then the total energy in the Universe). And that is not possible, it has to be limited at least by the total energy in the Universe. Or you could say that there can be an electron energy level (really existing in an atom), that is even lower, and lower and so on, into (almost) 0. But that would mean 0 frequency and an infinite wavelength (or simply just a wavelength longer then the scale of the universe).

4. But that (infinite wavelength) would mean zero frequency. And that is not possible even theoretically. And experimentally really not, why? Because of what Anna V was writing about the Planck-length and Planck-time. In the experiments, we can only measure things until a certain minimal size. And theoretically, usually we say that anything smaller then the Planck-length or smaller then the Planck-time does not mean anything. You understand, because of what I wrote about what the Planck-constant means. And the Planck-length and Planck time come from there. Not only are we not able to measure those frequencies, under the Planck-length or Planck-time, but theoretically they do not mean anything. Why? Because EM waves are information. And nothing can be smaller then (in this case the frequency of) information itself.

5. So if the minimal size of the frequency of the EM-waves, (the information )itself cannot be smaller then something (Planck-length or Planck-time) then there is nothing that could emit a photon with a frequency smaller then that (Planck-time) and a wavelength greater then that in our experiments.

6. Though, theoretically, there can be an atom in the universe somewhere that has an electron with an energy level that is so big that will give an emitted photon with energy so big, that(if divided by the Planck-constant) gives a frequency that would mean a wavelength of the size of the universe. So Josephus is right too, but I don't think there is a reason for talking about a wavelength larger then the size of the universe.