KAHIBARO
Table of Contents
Light carries momentum
In everyday mechanics, momentum is usually associated with moving matter. A ball, a car, or a falling stone has momentum because it has mass and velocity. In quantum physics, light also carries momentum, even though a photon has no rest mass. This is one of the important ideas that links light to particle behavior.
A photon is the quantum of electromagnetic radiation. If a photon has energy $E$, then its momentum $p$ is
$$
p = \frac{E}{c}
$$
where $c$ is the speed of light in vacuum. Using the relation between photon energy and frequency,
$$
E = hf
$$
we obtain
$$
p = \frac{hf}{c}
$$
Since wavelength $\lambda$ and frequency are related by $c = f\lambda$, this can also be written as
$$
p = \frac{h}{\lambda}
$$
This is the most common formula for photon momentum.
Important photon momentum formulas:
$$
p = \frac{E}{c}
$$
$$
p = \frac{hf}{c}
$$
$$
p = \frac{h}{\lambda}
$$
A shorter wavelength means a larger photon momentum.
Why this is surprising
In classical mechanics, momentum is often written as $p = mv$. That formula cannot be applied directly to photons, because photons do not have rest mass. Yet experiments show clearly that light can push on matter, transfer momentum, and change the motion of particles.
This means momentum is more fundamental than the simple expression $mv$. For photons, the correct relation is the one above, based on energy and wavelength.
Direction of photon momentum
Momentum is a vector quantity, so photon momentum has both magnitude and direction. A photon’s momentum points in the direction the light travels.
If a beam of light travels to the right, its photons carry momentum to the right. When light is absorbed or reflected by an object, momentum is transferred to that object.
Momentum transfer by absorption and reflection
If an object absorbs a photon, it receives the photon’s momentum.
If a photon with momentum $p$ is absorbed, the object gains momentum $p$ in the direction of the incoming light.
Reflection produces an even larger change in momentum. If a photon strikes a surface and reverses direction, its momentum changes from $+p$ to $-p$. The total change in the photon’s momentum is
$$
\Delta p = -p - (+p) = -2p
$$
So the surface receives momentum of magnitude $2p$.
This is why reflected light can exert a stronger push than absorbed light.
If light is absorbed, momentum transferred is
$$
\Delta p = p
$$
If light is reflected straight back, momentum transferred is
$$
\Delta p = 2p
$$
Photon momentum and radiation pressure
Because light carries momentum, a beam of light can exert pressure on a surface. This is called radiation pressure. Even though the effect is usually very small, it is real and measurable.
Examples include sunlight pushing on dust in space, laser beams exerting forces on tiny particles, and the idea of solar sails for spacecraft. In all of these cases, the physical origin of the force is momentum transfer from photons.
A stronger beam transfers more momentum per second, so it produces a greater force.
Comparing photon properties
The momentum of a photon depends on its wavelength, frequency, and energy. The table below shows the relationships.
| Property increases | What happens to photon momentum |
|---|---|
| Energy $E$ increases | Momentum increases |
| Frequency $f$ increases | Momentum increases |
| Wavelength $\lambda$ increases | Momentum decreases |
So high-frequency light, such as X rays, has more momentum per photon than low-frequency light, such as radio waves.
Simple examples
Consider a photon with wavelength $\lambda = 500 \text{ nm} = 5.0 \times 10^{-7}\text{ m}$. Its momentum is
$$
p = \frac{h}{\lambda}
$$
Using $h = 6.63 \times 10^{-34}\text{ J s}$,
$$
p = \frac{6.63 \times 10^{-34}}{5.0 \times 10^{-7}}
= 1.33 \times 10^{-27}\text{ kg m/s}
$$
This is a very small momentum for one photon, but a large number of photons in a beam can produce a noticeable effect.
As another example, suppose a photon has energy $E = 3.0 \times 10^{-19}\text{ J}$. Then
$$
p = \frac{E}{c}
= \frac{3.0 \times 10^{-19}}{3.0 \times 10^8}
= 1.0 \times 10^{-27}\text{ kg m/s}
$$
Connection to quantum physics
Photon momentum is a key quantum idea because it shows that light behaves like particles in collisions and interactions. Light does not just carry energy, it also carries momentum that can be exchanged with matter.
This idea helps explain many microscopic processes, especially when light interacts with electrons and atoms. It also leads naturally to the broader quantum relation between wavelength and momentum that later applies to matter particles as well.
A photon has no rest mass, but it still has momentum.
Its momentum is determined by its energy or wavelength:
$$
p = \frac{E}{c} = \frac{h}{\lambda}
$$
This momentum is transferred when light interacts with matter.