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 $u$:

$$E = hu$$

Using the speed of light $c = \lambda u$, 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

Symbol	Quantity	Value
$h$	Planck constant	$6.626 \times 10^{-34}\ \text{J·s}$
$c$	Speed of light in vacuum	$2.998 \times 10^{8}\ \text{m·s}^{-1}$
$\lambda$	Wavelength of photon	Variable
$u$	Frequency of photon	Variable

Derivation of Photon Momentum

1. Start from the wave relation $c = \lambda u$.
2. Combine with $E = hu$ to get $E = hc/\lambda$.
3. Use the relativistic relation $E^2 = (pc)^2 + (m_0c^2)^2$ with $m_0 = 0$ 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 $E = hu$ and momentum $p = h/\lambda$. These relationships link the wave and particle descriptions of light and underpin many phenomena in modern physics.