What is the relationship between the classical wavelength of light and the quantum mechanical wavefunction of a photon?

It is common to speak of the 'wavelength' of light. For instance visible light has a wavelength of around 400 to 700 nm. A single photon can also have a wavelength, given apparently by $$\lambda = \frac{hc}{E}$$. My understanding is that a photon's wavelength is not related to the physical size of a photon, which is a actually a point particle. Instead the wavelength of a photon is related to the probability of finding a photon at a particular coordinate in spacetime.

So is the wavelength of a photon from the quantum perspective the same value as the wavelength of a classical EM wave? For example, if a red traffic light is shining at 700nm, does each photon have 'quatum wavelength' of 700nm? Does this mean that (for a given instant in time) the probability of detecting said photon rises and falls every 700nm?

It would also be nice to see whavetever equations are relevant.

For example, if a red traffic light is shining at 700nm, does each photon have 'quatum wavelength' of 700nm

yes

Does this mean that (for a given instant in time) the probability of detecting said photon rises and falls every 700nm?

Yes, if the wave is plane polarized, so that the classical energy density has this oscillation. The probability density has to scale the same way as the classical energy density, so that the average rate of energy deposition on a detector is in accord with the correspondence principle.

• Thanks! This seems to contradict @nosuchthingasmagic 's answer. Any relevant equations or resources that could help me understand this better? Commented Nov 19, 2020 at 21:50

The question asks about the wavelength and the wavefunction. To clarify, those are two distinct things. The wavelength is a property (e.g., 700 nm), while the wavefunction is a function that can vary over space and time and gives the probability of detecting the photon in a certain state. While there are different theories, a photon has just one wavelength.

I remember being really impressed when I was in school that the probabilistic nature of light usually associated with quantum mechanics is already apparent from classical electricity and magnetism. To recap, in classical E&M, what is "waving" is the electromagnetic field. According to classical E&M, the density per unit volume stored in an electromagnetic field is proportional to $$\frac{1}{2} \left(\epsilon_0 E^2 + \frac{1}{\mu_0}B^2\right).$$ For a monochromatic plane wave, the electric and magnetic contributions are equal, so it is proportional to $$E^2$$. (You can find details in any standard source on E&M.)

Now if you accept that the field represents a group of photons in which each photon has energy $$\hbar\omega$$ (where $$\omega$$ is the frequency of the electromagnetic wave and is related to the wavelength $$\lambda$$ by $$\lambda= \frac{2\pi c}{\omega}$$), the density of photons in the field is proportional $$\frac{E^2}{\hbar \omega}$$, which you can think of as the probability density for finding a photon. With the $$E^2$$, you already get the interference effects of quantum mechanics.

I should also note that since photons are relativistic particles, you need to go beyond non-relativistic quantum mechanics for an accurate description (see, e.g.: What equation describes the wavefunction of a single photon?)

• This has been down voted but it seems the same as the answer by @user280073. Not that I wish the latter to be down voted too. Just a comment. Commented Nov 18, 2020 at 9:56
• Thanks. I read the question more carefully and tried to provide a more direct answer. Commented Nov 19, 2020 at 4:54
• Thanks @nosuchthingasmagic. So if understand you correctly, the lambda term for a monochromatic photon/EM wave would have the same value in both classical EM and quantum descriptions. On the other hand, this wavelength value is not the literal spacing over which a photon's probability density fluctuates (although it is proportional). And it seems like you're saying it's not the literal spacing of crests and troughs in classical EM theory either? (I'm assuming no interference with other phtons/waves) Commented Nov 19, 2020 at 21:41
• @Michael Yes, the lambda/wavelength is a unique property of monochromatic light/photons. As far as I am aware, there is no quantum wavelength that is distinct from an E&M wavelength. And yes, the wavelength is typically a parameter in the wavefunction, but the interesting effects occur when there is interference. For example, in the double slit experiment, the intensity depends on $\lambda$ but the oscillations on the screen are not separated by $\lambda$ itself. Commented Nov 21, 2020 at 3:58
• Ah I see. It's so hard to make precise statements without math. I'm not asking about the interference patterns. I'm imagining a single photon in vacuum with some lambda value. Now imagine a detector that moves towards the photon source. The probability of detecting a photon will rise and fall as the detector moves towards the source right? How is the rate of rising and falling probability related to lambda? Thanks again Commented Nov 21, 2020 at 17:09