29.4 Photon Momentum

Photons carry both energy and momentum, even though they have no mass. Understanding photon momentum is key to explaining how light interacts with matter at the quantum level, from X-ray scattering to the photoelectric effect.

The Compton effect is one of the strongest pieces of evidence that photons behave like particles. When a photon collides with an electron, it transfers momentum and energy, just like a collision between two physical objects. This section covers the math behind photon momentum, how Compton scattering works, and why conservation laws still apply at the quantum scale.

Photon Momentum and Interactions

Photon momentum relationships

Unlike massive objects (where $p = mv$), photons have zero mass but still carry momentum. Photon momentum is directly proportional to energy and inversely proportional to wavelength:

$p = \frac{E}{c} = \frac{h}{\lambda}$

• $p$ = photon momentum
• $E$ = photon energy
• $c$ = speed of light ($3 \times 10^8$ m/s)
• $h$ = Planck's constant ($6.626 \times 10^{-34}$ J·s)
• $\lambda$ = photon wavelength

Because momentum and wavelength are inversely related, shorter-wavelength photons pack more momentum. X-ray photons (wavelengths around $10^{-10}$ m) carry far more momentum than radio wave photons (wavelengths of meters or longer). Similarly, ultraviolet photons have more momentum than infrared photons because their wavelengths are shorter.

This is why high-energy photons like X-rays and gamma rays can knock electrons around, while low-energy radio waves pass through matter with barely any interaction.

Compton effect significance

The Compton effect is the scattering of a photon by a charged particle, usually an electron. Here's what happens during a Compton scattering event:

1. An incoming photon (typically an X-ray) strikes an electron that is roughly at rest.
2. The photon transfers some of its energy and momentum to the electron.
3. The photon scatters off at some angle $\theta$ with a longer wavelength (lower energy) than it started with.
4. The electron recoils, carrying away the energy and momentum the photon lost.

Think of it like a cue ball hitting a stationary billiard ball. The cue ball slows down (loses energy), and the target ball moves off with the transferred momentum.

The shift in the photon's wavelength depends on the scattering angle and is given by the Compton wavelength shift equation:

$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$

• $m_e$ = mass of the electron ($9.11 \times 10^{-31}$ kg)
• $\theta$ = scattering angle (the angle between the incoming and scattered photon directions)
• $\frac{h}{m_e c}$ = the Compton wavelength of the electron, approximately $2.43 \times 10^{-12}$ m

A few things to notice about this equation. When $\theta = 0°$, $\cos\theta = 1$, so $\Delta\lambda = 0$: no scattering means no wavelength shift. When $\theta = 180°$ (a head-on backscatter), $\cos\theta = -1$, and the wavelength shift is at its maximum value of $\frac{2h}{m_e c}$. Larger scattering angles always produce larger wavelength shifts.

The Compton effect was historically important because classical wave theory could not explain why the scattered light's wavelength would change. Only by treating light as particles (photons) with definite momentum could the results be predicted. This was strong evidence for the quantum nature of light.

Conservation in photon interactions

The same conservation laws you've used for collisions between macroscopic objects apply here. In any photon-particle interaction, both total momentum and total energy are conserved.

For Compton scattering, the momentum conservation equation is:

$\vec{p}_{\text{photon,initial}} + \vec{p}_{\text{electron,initial}} = \vec{p}_{\text{photon,final}} + \vec{p}_{\text{electron,final}}$

The electron is typically at rest before the collision, so its initial momentum is zero. This simplifies the problem: all the initial momentum comes from the incoming photon.

Because momentum is a vector, you need to apply conservation separately in two directions (usually along the incoming photon's path and perpendicular to it). By combining momentum conservation with energy conservation, you can:

• Derive the Compton wavelength shift equation shown above
• Calculate the kinetic energy of the recoiling electron for a given scattering angle
• Predict the direction the electron recoils

For example, if you know the incoming photon's wavelength and the scattering angle, you can find the scattered photon's wavelength using the Compton equation, then use energy conservation to determine how much kinetic energy the electron gained.

These conservation principles aren't limited to Compton scattering. They apply to all photon interactions with matter, including the photoelectric effect and photon absorption.

Wave-particle duality and related concepts

Photon momentum connects to a broader idea: wave-particle duality. Photons behave like waves in some experiments (interference, diffraction) and like particles in others (photoelectric effect, Compton scattering).

The de Broglie wavelength extends this duality to all matter. Any particle with momentum $p$ has an associated wavelength:

$\lambda = \frac{h}{p}$

This is the same equation as $p = \frac{h}{\lambda}$ rearranged. For photons, it gives the familiar relationship between wavelength and momentum. For massive particles like electrons, it predicts that fast-moving particles have very short wavelengths, which has been confirmed by electron diffraction experiments.

The photoelectric effect is another phenomenon that demonstrates light's particle nature. When photons strike a metal surface, they can eject electrons, but only if each individual photon has enough energy. This can't be explained by treating light purely as a wave. Together, the photoelectric effect and Compton scattering built the case that light is quantized into photons with definite energy and momentum.