# Spectral decomposition of radio waves and quantization

Background: I was taught that all electromagnetic radiation can be thought of as a sine wave, and that what we receive on a radio, for example, is actually the sum of sine waves over a range of frequencies. We can decompose a non-sinusoidal signal wave into its sine components and back via the Fourier transform.

I was also taught that all electromagnetic radiation can be quantized into photons with a particular frequency.

Question context: The first view of light as a continuous wave allows a continuity of the spectral decomposition. However the second view seems to indicate that this continuous spectrum must be discretized.

Question: How are the specific discretization of the power spectrum accomplished? If someone were broadcasting a signal with an infinite spectral decomposition (eg a square wave), would a receiver set up to detect individual photons detect photons of any frequency with a probability equal to the magnitude of the spectrum at that frequency?

Assuming that's the case, if we imagine individual photons being emitted from the transmitter, does each photon carry the probability of being detected over the full range of the signal, or does each photon have a very narrow range of uncertainty in its frequency? Does this narrowness depend on the rate at which photons are being emitted?

• Taking the limit to infinite frequency means infinite energy for the photons. That sounds like UV divergence or at least something to be careful about. For everyday radio waves the number of photons is so high that quantization does not become too apparent. Perhaps you want to look into the thermal spectrum and Planck's UV catastrophe paradox as well. Commented Oct 2, 2016 at 19:55
• Possible duplicates: physics.stackexchange.com/q/73959/2451 and links therein. Commented Oct 2, 2016 at 20:38

## 1 Answer

Note that the photon picture says that the intensity of an electromagnetic wave is discretized. It doesn't say the frequency is discretized. We can still have photons of any frequency (assuming free space boundary conditions. If we put on different boundary conditions [such as putting the field in a cavity] we might find the frequency is discretized as well, but in this question we're talking about RF broadcasting so we'll assume free space.)

Question: How are the specific discretization of the power spectrum accomplished? If someone were broadcasting a signal with an infinite spectral decomposition (eg a square wave), would a receiver set up to detect individual photons detect photons of any frequency with a probability equal to the magnitude of the spectrum at that frequency?

That's exactly right.

Assuming that's the case, if we imagine individual photons being emitted from the transmitter, does each photon carry the probability of being detected over the full range of the signal, or does each photon have a very narrow range of uncertainty in its frequency? Does this narrowness depend on the rate at which photons are being emitted?

You have to be careful in thinking about individual photons being emitted. I think your first statement is more correct that "each photon carries the probability of being detected over the full range of the signal." I think the more correct statement is this: The state emitted from the transmitter is a superposition state consisting of photons with frequencies all across the spectrum of the signal. If you want to say we are emitting one photon you can say that we are emitting a signal where the expectation value of the number of photons measured is one, but the state is still going to be a superposition of many states of different frequencies.

Here's another question which gets at the heart of what we mean by "individual photons". Say I produce a state which is a superpostion of one photon at 100 MHz and one photon at 200 MHz. Does this state have one or two photons in it? I think the right answer is one photon. If you measure the state you'll always find one photon.