understand that a photon is a quantum of electromagnetic energy

Energy and Momentum of a Photon

Learning Objective

Understand that a photon is a quantum of electromagnetic energy and be able to calculate its energy and momentum.

Key Concepts

• Photon as a particle of light.
• Quantisation of electromagnetic radiation.
• Relationship between frequency, wavelength, energy and momentum.

Fundamental Relations

The energy \$E\$ of a photon is directly proportional to its frequency \$\nu\$:

\$E = h\nu\$

Using the speed of light \$c = \lambda \nu\$, the energy can also be expressed in terms of wavelength \$\lambda\$:

\$E = \frac{hc}{\lambda}\$

The momentum \$p\$ of a photon is related to its wavelength by:

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

Constants

Derivation of Photon Momentum

1. Start from the wave relation \$c = \lambda \nu\$.
2. Combine with \$E = h\nu\$ to get \$E = hc/\lambda\$.
3. Use the relativistic relation \$E^2 = (pc)^2 + (m0c^2)^2\$ with \$m0 = 0\$ for a photon, giving \$E = pc\$.
4. Therefore \$p = E/c = h/\lambda\$.

Applications

• Photoelectric effect – photons eject electrons from metal surfaces.
• Radiation pressure – momentum transfer from light to surfaces.
• Compton scattering – change in photon wavelength due to collision with electrons.

Example Problem

Problem: Calculate the energy and momentum of a photon with wavelength \$500\ \text{nm}\$ (green light).

Solution:

1. Convert wavelength to metres: \$\lambda = 500\ \text{nm} = 5.00 \times 10^{-7}\ \text{m}\$.
2. Energy:

   \$E = \frac{hc}{\lambda} = \frac{(6.626 \times 10^{-34})(2.998 \times 10^{8})}{5.00 \times 10^{-7}} \approx 3.98 \times 10^{-19}\ \text{J}\$

3. Momentum:

   \$p = \frac{h}{\lambda} = \frac{6.626 \times 10^{-34}}{5.00 \times 10^{-7}} \approx 1.33 \times 10^{-27}\ \text{kg·m·s}^{-1}\$

Suggested Diagram

Summary

A photon carries quantised energy \$E = h\nu\$ and momentum \$p = h/\lambda\$. These relationships link the wave and particle descriptions of light and underpin many phenomena in modern physics.