recall and use E = hf

Energy and Momentum of a Photon (Cambridge IGCSE/A‑Level 9702)

Learning Objective (AO1 + AO2)

Recall the fundamental photon relations and use them to calculate photon energy, momentum and related quantities in a range of contexts.

• \(E = hf\)
• \(E = \dfrac{hc}{\lambda}\)
• \(p = \dfrac{E}{c} = \dfrac{h}{\lambda} = \dfrac{hf}{c}\)

Key Concepts

• Photons are quanta of electromagnetic radiation. They have zero rest mass but carry energy and momentum.
• Energy–frequency relation: \[ E = hf \qquad\text{with } h = 6.626\times10^{-34}\ \text{J·s} \]
• Using the wave relation \(c = \lambda f\) the same energy can be written in terms of wavelength: \[ E = \frac{hc}{\lambda} \]
• For a mass‑less particle \(E = pc\); therefore photon momentum is \[ p = \frac{E}{c}= \frac{h}{\lambda}= \frac{hf}{c} \]
• Photon momentum, although tiny, produces measurable effects such as radiation pressure, the photo‑electric effect, and Compton scattering.

Useful Constants

Constant	Symbol	Value	Units
Planck constant	\(h\)	6.626 × 10⁻³⁴	J·s
Speed of light in vacuum	\(c\)	2.998 × 10⁸	m·s⁻¹
Elementary charge	\(e\)	1.602 × 10⁻¹⁹	C
Electron rest mass	\(m_e\)	9.109 × 10⁻³¹	kg

Typical Photon Energies

Radiation type	Wavelength \(\lambda\) (nm)	Frequency \(f\) (THz)	Energy \(E\) (eV)
Radio (AM)	≈ 10⁶	≈ 0.3	≈ 1.2 × 10⁻⁶
Microwave (2.45 GHz)	≈ 122 000	2.45	≈ 1.0 × 10⁻⁵
Infrared (10 µm)	10 000	30	≈ 0.124
Visible (green, 550 nm)	550	545	≈ 2.25
Ultraviolet (200 nm)	200	1500	≈ 6.20
X‑ray (0.1 nm)	0.1	3 × 10⁶	≈ 12.4 × 10³
Gamma (0.01 nm)	0.01	3 × 10⁷	≈ 124 × 10³

Derivation of Photon Momentum

Starting from the energy–frequency relation and the wave‑speed definition:

\[ E = hf,\qquad c = \lambda f \]

Eliminate \(f\) to obtain the wavelength form of the energy:

\[ E = h\frac{c}{\lambda}= \frac{hc}{\lambda} \]

For a particle with zero rest mass the relativistic relation \(E = pc\) holds, giving

\[ p = \frac{E}{c}= \frac{h}{\lambda}= \frac{hf}{c} \]

Photon‑Related Phenomena (AO2)

Photo‑electric Effect

• Maximum kinetic energy of ejected electrons: \[ K_{\max}=hf-\phi \]
• Threshold frequency: \[ f_{\text{thr}}=\frac{\phi}{h} \]
• Both equations are used to determine the work function \(\phi\) or the kinetic energy for a given frequency.

Compton Scattering

The change in photon wavelength after scattering from a free electron is

\[ \Delta\lambda = \frac{h}{m_ec}(1-\cos\theta) \]

derived from conservation of energy (\(hf\)) and momentum (\(h/\lambda\)).

Radiation Pressure

• Absorbing surface: \(F = \dfrac{P}{c}\)
• Perfectly reflecting surface: \(F = \dfrac{2P}{c}\) (momentum reversal doubles the transfer).

de Broglie Wavelength (Matter Waves)

Any particle of momentum \(p\) has an associated wavelength

\[ \lambda_{\text{dB}} = \frac{h}{p} \]

For a non‑relativistic particle with kinetic energy \(K\), \(p = \sqrt{2mK}\) and

\[ \lambda_{\text{dB}} = \frac{h}{\sqrt{2mK}} \]

Example: an electron with \(K = 100\ \text{eV}\) has

\[ \lambda_{\text{dB}} = \frac{6.626\times10^{-34}}{\sqrt{2(9.109\times10^{-31})(100\times1.602\times10^{-19})}} \approx 1.23\times10^{-10}\ \text{m} \;(0.123\ \text{nm}) \]

Energy Levels & Line Spectra

• When an electron jumps between two atomic energy levels \(E_1\) and \(E_2\), a photon is emitted or absorbed with energy \[ hf = |E_1-E_2| \]
• Balmer series (visible hydrogen lines) – e.g. the \(H_\alpha\) line at \(\lambda = 656.3\ \text{nm}\) corresponds to \[ \Delta E = \frac{hc}{\lambda}= \frac{(6.626\times10^{-34})(2.998\times10^{8})}{656.3\times10^{-9}} \approx 3.03\times10^{-19}\ \text{J}=1.89\ \text{eV} \]

Worked Example Problems (AO2)

1. Energy of a green photon – \(\lambda = 550\ \text{nm}\) \[ E = \frac{hc}{\lambda}= \frac{(6.626\times10^{-34})(2.998\times10^{8})}{550\times10^{-9}} = 3.61\times10^{-19}\ \text{J}=2.25\ \text{eV} \]
2. Momentum of an X‑ray photon – \(\lambda = 0.1\ \text{nm}\) \[ p = \frac{h}{\lambda}= \frac{6.626\times10^{-34}}{0.1\times10^{-9}} = 6.63\times10^{-24}\ \text{kg·m·s}^{-1} \]
3. Radiation pressure on an absorbing surface – laser power \(P = 5\ \text{W}\) \[ F = \frac{P}{c}= \frac{5}{2.998\times10^{8}} = 1.67\times10^{-8}\ \text{N} \]
4. Radiation pressure on a reflecting surface – same 5 W laser \[ F = \frac{2P}{c}= 3.34\times10^{-8}\ \text{N} \]
5. Photo‑electric threshold frequency – \(\phi = 2.0\ \text{eV}\) \[ f_{\text{thr}} = \frac{\phi}{h}= \frac{2.0\times1.602\times10^{-19}}{6.626\times10^{-34}} = 4.8\times10^{14}\ \text{Hz} \]
6. Maximum kinetic energy for a given frequency – \(\lambda = 250\ \text{nm}\) (so \(f = c/\lambda\)) and \(\phi = 1.5\ \text{eV}\) \[ f = \frac{c}{\lambda}= \frac{2.998\times10^{8}}{250\times10^{-9}}=1.20\times10^{15}\ \text{Hz} \] \[ K_{\max}=hf-\phi = (6.626\times10^{-34})(1.20\times10^{15})-1.5\,\text{eV} = 4.96\times10^{-19}\ \text{J}-1.5\,\text{eV}=2.6\ \text{eV} \]
7. Compton wavelength shift – incident \(\lambda = 0.071\ \text{nm}\), \(\theta = 60^{\circ}\) \[ \Delta\lambda = \frac{h}{m_ec}(1-\cos60^{\circ}) = 2.43\times10^{-12}\,(1-0.5)=1.22\times10^{-12}\ \text{m} \] \[ \lambda' = 0.071\ \text{nm}+0.00122\ \text{nm}=0.0722\ \text{nm} \]
8. de Broglie wavelength of a 100 eV electron (see de Broglie section) – \(\lambda_{\text{dB}} = 0.123\ \text{nm}\).
9. Energy difference for the H\(_\alpha\) line – \(\lambda = 656.3\ \text{nm}\) gives \(\Delta E = 1.89\ \text{eV}\).

Common Misconceptions

• Photons have zero **rest** mass, yet they carry momentum because \(p = h/\lambda\) does not involve mass.
• The relation \(E = hf\) is universal for all electromagnetic radiation, not just visible light.
• When converting between wavelength and frequency, keep units consistent (nm → m, THz → Hz).
• Radiation pressure on a mirror is twice that on an absorbing surface; the factor of 2 is often omitted.
• De Broglie wavelength applies to *matter* particles, not to photons (which already have \(\lambda = h/p\)).
• Photon energy equals the *difference* between atomic energy levels, not the absolute value of a single level.

Suggested Classroom Activities

1. Calculate photon energies for the seven colours of the visible spectrum and plot \(E\) versus \(\lambda\). Discuss the linear relationship between \(E\) and \(1/\lambda\).
2. Demonstrate radiation pressure using a light‑mill or a small solar‑sail model; compare measured forces with the predictions \(F = P/c\) (absorbing) and \(F = 2P/c\) (reflecting).
3. Measure the wavelength of a He‑Ne laser with a diffraction grating, then compute its momentum and the corresponding radiation pressure on a suspended mirror.
4. Perform a photo‑electric experiment (metal cathode + variable‑frequency LED) to verify the threshold frequency and the equation \(K_{\max}=hf-\phi\).
5. Use a computer simulation of Compton scattering to visualise the wavelength shift and confirm energy‑momentum conservation.
6. Calculate de Broglie wavelengths for electrons, neutrons and larger particles (e.g., C\(_{60}\) molecules) and discuss why wave behaviour becomes unobservable for macroscopic masses.
7. Analyse a spectral line (e.g., the Balmer series) by converting its wavelength to the energy difference between atomic levels.