The blank the wavelength of a photon, the blank its momentum is. The blank the frequency of a photon, the blank its momentum is. (a) Longer, larger, higher, larger. (b) Shorter, larger, lower, larger. (c) Shorter, smaller, higher, smaller. (d) Longer, larger, higher, smaller. (e) Shorter, larger, higher, larger.
This question is asking us for the qualitative relationship between the wavelength and frequency of a photon and its momentum. If we don’t remember these qualitative relationships offhand, we can figure them out from some quantitative relationships. The two quantitative relationships we need express momentum as the Planck constant divided by the wavelength of a photon or the Planck constant times the frequency of a photon divided by the speed of light. We can actually get one of these relationships from the other by recalling that the speed of light is the wavelength of a photon times its frequency.
Anyway, looking back at the formulas, we see that the formula that relates wavelength and momentum has the general form one quantity is equal to a constant divided by the other quantity, which is the general form of an inversely proportional relationship. In our other formula relating momentum and frequency, note that ℎ is a constant and 𝑐 is also a constant. So, ℎ divided by 𝑐 is a constant. And this relationship has the general form one quantity is equal to a constant times the other quantity. This is the general form of a directly proportional relationship. So, we know that momentum and wavelength are inversely proportional and momentum and frequency are directly proportional.
So, let’s understand the qualitative features of these kinds of relationships. When two quantities are inversely proportional, when the magnitude of one increases, the magnitude of the other decreases and vice versa. This is why it is called inversely proportional because the magnitudes change oppositely. We can also see this from the form of their relationship. With the wavelength in the denominator, we can see that a larger wavelength means a fraction with a larger denominator. But the larger the denominator of a fraction, the smaller the value of the fraction. Similarly, the smaller the denominator of a fraction, the larger the value of the fraction.
So, the longer the wavelength, the smaller the momentum, and the shorter the wavelength, the larger the momentum. Of our answer choices, (a) and (d) suggest that momentum is larger for longer wavelengths, and (c) suggests that momentum is smaller for shorter wavelengths. But neither of these is correct for an inversely proportional relationship. So, only (b) and (e) are possible correct answers.
To figure out which one is correct, we now need to consider directly proportional relationships. In a directly proportional relationship, as its name suggests, as the magnitude of one quantity increases, the magnitude of the other quantity increases as well. And similarly, as the magnitude of one quantity decreases, the magnitude of the other also decreases. So, higher-frequency photons have larger momenta, and lower-frequency photons have smaller momenta. Of our remaining answer choices, only (e) higher frequency and larger momentum correctly shows a directly proportional relationship. So, the correct answer is (e). The shorter the wavelength of a photon, the larger its momentum is, and the higher the frequency of a photon, the larger its momentum is.