In this explainer, we will learn how to calculate the momentum of a photon given its frequency or wavelength.

Recall that there are some physical phenomena involving light that are best described using a wave model. Such phenomena include refraction and diffraction. Other phenomena are best described using a particle model of light. Such phenomena include the photoelectric effect.

In the wave model of light, light has a wavelength and a frequency. The wavelength of a wave is the distance between any two corresponding points on the wave, as shown in the diagram below. The frequency of a wave is the number of cycles of the wave that pass a point each second.

Recall that if the wavelength of a wave is and the frequency of the wave is , then the speed of the wave, , is given by

Recall also that, in free space, light travels at a speed of approximately m/s. This constant is given the label , so for a light wave,

In the particle model of light, the energy of light is divided into
โpacketsโ of energy, called **photons**. Photons can be absorbed or
emitted by atoms. The photoelectric effect is when an electron in an atom
absorbs a photon, is ejected from the atom, and leaves the material that the
atom is part of entirely.

Recall that the energy of a single photon is related to the frequency of the wave that describes the light. The energy, , of a photon is given by where is the Planck constant, and has a value of Jโ s.

Because the frequency and wavelength of a light wave are related by , we can also express the energy of a photon in terms of the wavelength of the wave:

In addition to being discrete โpacketsโ of energy, photons also
have *momentum* and as such can *exert a force*. This might seem
counterintuitive at first, as in our everyday interactions with light, we do
not *feel* it exerting a force. If you hold your hand up to a light bulb,
you do not feel a force on your hand due to the light.

This might further seem unintuitive because momentum is usually calculated
using the formula . For an object with mass
moving at speed
, its momentum, , is the product of its
mass and its speed. But photons have **zero mass**. If ,
then . Following this formula, if something has
zero mass, then it also must have zero momentum, regardless of its
speed.

However, there are limitations to when can be used. The formula
cannot be used for objects that are moving very
fastโobjects moving close to the speed of light. This alone means that the
formula cannot be used for photons, which, of course, move *at* the speed
of light. The formula also cannot be used for massless particles,
which includes photons.

The momentum of a photon is instead related to its *wavelength* and can be
calculated using the formula
where is the momentum of the photon and
is the Planck constant, as before.

Notice that the momentum of a photon is **inversely proportional** to its
wavelength. This means that as the wavelength of a photon *increases*, or
as the light *becomes redder*, its momentum *decreases*.

We can use this formula to see why we do not experience the momentum of photons in our everyday interactions with light. Consider a photon of red light, which has a wavelength of about 700 nm, or m. The momentum of the photon is

Units of joule-seconds per metre (Jโ s/m) are equivalent to units of kilogram-metres per second (kgโ m/s), so the momentum of the photon is kgโ m/s.

This is a *very* small value. A single photon does not have much
momentum. Even if we consider the total momentum of all of the photons emitted
by a light bulb each second, it is still a very small value. A
100 W light bulb emits approximately
photons each second. The total momentum of all of
these photons is

This is a very small value of momentum, and this is why we do not perceive the momentum of light in our everyday interactions with it.

The momentum of photons becomes very important, however, when we deal with the
interactions between photons and other particles, such as electrons.
Photonsโparticularly high-energy photons, such as x-ray photonsโ*can*
impart a significant momentum onto other particles.

### Formula: The Momentum of a Photon in terms of Its Wavelength

The momentum, , of a photon is equal to the Planck constant, , divided by the wavelength, , of the photon:

Because, for light, wavelength and frequency are related by , we can also express the momentum of a photon in terms of its frequency. If first we rearrange to make the subject, we get

Substituting this into the formula for the momentum of a photon, we get

### Formula: The Momentum of a Photon in terms of Its Frequency

The momentum, , of a photon is equal to the Planck constant, , multiplied by the frequency, , of the photon, divided by the speed of light, :

### Example 1: Calculating the Momentum of a Photon given Its Wavelength

What is the momentum of a photon that has a wavelength of 500 nm? Use a value of Jโ s for the Planck constant. Give your answer in scientific notation to two decimal places.

### Answer

We can use the formula to work out the momentum, , of the photon, where is the Planck constant, and is the wavelength of the photon.

First, letโs convert the value we have been given for the wavelength into metres. Recall that , so .

Now we can substitute this value, as well as the value for the Planck constant given in the question, into the above formula. Doing this, we get

Units of joule-seconds per metre (Jโ s/m) are equivalent to units of kilogram-metres per second (kgโ m/s). The question tells us to give our answer to 2 decimal places, so our final answer is

### Example 2: Calculating the Momentum of a Photon given Its Frequency

A low-frequency radio wave has a frequency of 200 kHz. What is the momentum of a radio-wave photon with this frequency? Use a value of Jโ s for the Planck constant. Give your answer in scientific notation to two decimal places.

### Answer

We can use the formula to find the momentum, , of the photon, where is the Planck constant, is the frequency of the photon, and is the speed of light.

First, letโs convert the value we have been given for the frequency into hertz. Recall that , so .

Now we can substitute this value, as well as the value for the Planck constant given in the question, into the above formula. We can use a value of m/s for the speed of light. Doing this, we get

Recall that units of hertz are equal to units of 1/s, so

Units of joule-seconds per metre (Jโ s/m) are equal to units of kilogram-metres per second (kgโ m/s), so the momentum of the photon is kgโ m/s.

In a scenario where we have identical photons, if we
know the wavelength of the photons, we can find the
**total** momentum of the photons using the formula

Alternatively, if we know the frequency of the photons, we can find the total momentum of the photons using the formula

### Example 3: Calculating the Total Momentum of Many Identical Photons

A laser produces photons, each with a frequency of Hz. What magnitude of momentum does producing these photons impart on the laser? Use a value of Jโ s for the Planck constant. Give your answer to three decimal places.

### Answer

We have been asked to find the magnitude of momentum
imparted on the laser by the photons as they are
emitted. Due to the conservation of momentum, the
change in the momentum of the laser will have the same
magnitude, but an opposite direction, to the change in
the momentum of the photons. Since the photons are
being *produced* by the laser, the change in
momentum of the photons is just the total momentum of
the photons.

The photons all have the same frequency, so they are identical. We can therefore use the formula to find the total momentum, , of the photons, where is the number of photons, is the Planck constant, is the frequency of the photons, and is the speed of light.

Substituting in the values given in the question, and using a value of m/s for the speed of light, we get

Recall that units of hertz are equal to units of 1/s, so

Units of joule-seconds per metre (Jโ s/m) are equal to units of kilogram-metres per second (kgโ m/s), so the total momentum of the photons is 3.757 kgโ m/s.

A momentum of this size *would* be perceptible.
However, the total energy of these photons is over
1 GJ. Even the most powerful lasers in the
world would take a long time to output this much
energy, so the change in speed of the laser would be
very slow.

### Key Points

- Photons have momentum, even though they have no mass.
- The momentum of a photon is directly proportional to its frequency and inversely proportional to its wavelength.
- If the wavelength of a photon is known, its momentum can be calculated using the formula
- If the frequency of a photon is known, its momentum can be calculated using the formula
- The total momentum of identical photons can be calculated using or