Write an expression for the momentum of a photon, given that it has energy 𝐸 and 𝑐 is the speed of light in a vacuum.
To start off, we can recall a mathematical relationship describing photon momentum. This relationship tells us that photon momentum — we’ll call it lowercase 𝑝 — is equal to Planck’s constant, ℎ, divided by the wavelength of the photon, 𝜆. Since we’re working with photons, which can be treated as particles, we can recall another relationship, this time for particle energy.
The energy, capital 𝐸, of a particle is equal to Planck’s constant multiplied by the frequency of oscillation of that particle. Typically, this relationship is used for very small particles where that frequency of oscillation is quite high. Looking at these two relationships, we see that they’re connected by Planck’s constant, ℎ, in these two equations. This means that if we want to, we can combine these two relationships.
We can do this by rearranging our photon momentum equation so that it reads 𝑝 multiplied by 𝜆 is equal to ℎ. And then considering our equation for particle energy, we see that ℎ is also equal to the particle energy, 𝐸, divided by its frequency. And therefore, we can see that the momentum of a photon multiplied by its wavelength is equal to its energy divided by its frequency.
Now what if we divide both sides of this equation by the wavelength of our photon, 𝜆? In that case, on the left-hand side of our expression, wavelength cancels. And then on the right-hand side, we now have it in our denominator. As we consider the frequency of our photon multiplied by its wavelength, this may bring to mind a relationship for wave speed. In general, the speed of a wave — we can call it 𝑣 — is equal to the wavelength of that wave times its frequency.
Even though we’re working with a photon, an individual packet of light, photons still exhibit wave-like properties. And therefore, this relationship applies to them. For a photon though, what’s the speed of that wave? It’s equal to 𝑐, the speed of light in a vacuum. This means we can replace 𝑓 times 𝜆 with 𝑐 in our equation. And this gives us our final expression for the momentum of a photon. It’s equal to the energy of the photon divided by the speed of light in vacuum.