### Video Transcript

The momentum of a photon is blank to its wavelength and blank to its frequency. Inversely proportional, inversely proportional; proportional, proportional; inversely proportional, proportional; proportional, inversely proportional; inversely proportional, equal.

We’re attempting to relate the momentum of a photon to its wavelength and frequency. And as we can see from the answer choices, the types of relationships we will be identifying are inversely proportional or proportional relationships.

To start, we’ll need to understand how to identify an inversely proportional or a proportional relationship. A proportional relationship, also sometimes called a directly proportional relationship to avoid confusion, between two quantities 𝐴 and 𝐵 can always be written as 𝐴 is equal to some constant times 𝐵. We call such a relationship directly proportional because the right-hand side is a product. So, when the magnitude of 𝐵 increases, the magnitude of the product increases as well, and so the magnitude of 𝐴 increases. Similarly, as the magnitude of 𝐵 decreases, the magnitude of 𝐴 decreases as well. So the magnitudes of 𝐴 and 𝐵 are directly related.

An inversely proportional relationship displays the opposite behavior to a directly proportional relationship. Instead of having a product, we have a fraction. That is, the quantity 𝐴 is equal to some constant divided by the quantity 𝐵. Because the quantity 𝐵 is in the denominator of the fraction, as the magnitude of 𝐵 increases, the value of the fraction decreases. So the magnitude of 𝐴 decreases as well. And vice versa, as the magnitude of 𝐵 decreases, the magnitude of 𝐴 increases. So the magnitudes of 𝐴 and B vary inversely.

To use these definitions to identify the relationships among physical quantities, we can write down a formula involving both of those quantities and then match it to either the form of a directly proportional relationship or the form of an inversely proportional relationship. The quantities that we will need to find formulas for are momentum and wavelength and momentum and frequency.

Let’s start with momentum and wavelength because the momentum of a photon is defined using the de Broglie relationship in terms of its wavelength. The de Broglie relationship tells us that 𝑝, the momentum of a photon, is defined as ℎ, the Planck constant, divided by 𝜆, the wavelength of the photon. 𝑝 and 𝜆 are the quantities that we want to relate, and the Planck constant is of course a constant. So this relationship has the form of one quantity is equal to a constant divided by another quantity. And this is the form of an inversely proportional relationship. So a photon’s momentum is inversely proportional to its wavelength.

To find a relationship between the momentum of a photon and its frequency, we can actually modify the de Broglie relation by replacing the wavelength with an appropriate function of frequency. All we need to do is recall that the speed of light, a constant, is equal to the frequency of a photon times its wavelength. If we divide both sides of this equation by the frequency, we can isolate the wavelength as the wavelength of a photon is equal to the speed of light divided by that photon’s frequency. Substituting this relationship into the de Broglie relationship, we have that 𝑝 is equal to ℎ divided by 𝑐 divided by 𝑓. But dividing by a fraction is the same as multiplying by that fraction’s reciprocal. So we can rewrite this as the Planck constant times the frequency of the photon divided by the speed of light.

Now, the Planck constant is a constant and the speed of light is a constant. So the Planck constant divided by the speed of light is also a constant. So now we have that the momentum is equal to a constant times the frequency. But if one quantity is a constant times the other quantity, then those two quantities are directly proportional. And this completes our statement.

Using the de Broglie relationship defining the momentum of a photon in terms of its wavelength and then modifying this relationship to give us a formula relating momentum and frequency, we have found the momentum of a photon is inversely proportional to its wavelength and proportional to its frequency.