Energy & Momentum of a Photon: Equation & Calculations

Light as a Particle
Light is a wave, but that’s not all it is. Light acting like a wave was relatively easy to produce. Even in the middle of the 19th century we knew that light could reflect, refract, and diffract. And, as far as we could tell, those effects could only be explained by light being a wave. But then, Albert Einstein noticed something. He showed in his photoelectric effect experiment that light could also behave as a particle.
In that experiment, he shone light onto some metal and found that electrons were ejected. When the light was brighter, more electrons were ejected, but those electrons didn’t move any faster – they didn’t have any extra energy. However, if he increased the energy of the light, making the light bluer, the electrons did move faster. Suffice it to say, this was not what he expected, and these observations led him to the realization that light must behave both as a wave and a particle; this is called wave-particle duality.
Energy of a Photon
It turns out that light contains energy in discrete packets (or particles) called photons. The amount of energy in those photons is calculated by this equation, E = hf, where E is the energy of the photon in Joules; h is Planck’s constant, which is always 6.63 * 10^-34 Joule seconds; and f is the frequency of the light in hertz.
The electrons in the metal were being hit by these photons, which gave them the energy needed to escape. Bluer light has more energy because it has a higher frequency, so the electrons that escaped were moving faster. Brighter light contained more photons, so more electrons left the metal, but those electrons were no faster than with the dimmer light because they could only be hit by one photon of light at a time.
This idea of electrons colliding with photons of light is why these observations can only be explained by treating light as a particle, rather than a wave. A wave can’t collide with anything, so if light was purely a wave, then brighter light should lead to higher energy electrons.
Momentum of a Photon
If light contains particles called photons, perhaps they should have momentum like any other particle. In fact, light is both a wave and a particle. So, not only does it have a momentum, it also has a wavelength. We relate these two quantities using something called the de Broglie wavelength: p = h / lambda. This equation says that the momentum of a photon, p, measured in kilogram meters per second, is equal to Planck’s constant, h, divided by the de Broglie wavelength of the light, lambda, measured in meters.
Example Calculation
So, how do we use these equations? Let’s try an example calculation:
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