Describe photons and calculate photon energy
Model electromagnetic radiation as photons and use E = hf to calculate the energy of one photon.
Section 22 · Quantum physics
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 22. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use wave frequency and wavelength, energy, momentum, SI prefixes, electronvolts and scientific notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In the photon model, what is one photon?
What you will learn
- Recognise the particulate nature of electromagnetic radiation
- Describe a photon as a quantum of electromagnetic energy
- Use E = hf and relate photon energy to frequency
A photon is a discrete packet of electromagnetic energy
Electromagnetic radiation can show wave behaviour, such as interference, and particulate behaviour. In the photon model, radiation is transferred in discrete packets called photons; one photon is a quantum, or smallest discrete packet, of electromagnetic energy for its frequency.
The energy of one photon is E = hf, where h is the Planck constant and f is the radiation frequency. Use h = 6.63 × 10⁻³⁴ J s when energy is required in joules. A photon of higher frequency has greater energy; changing the number of photons changes total energy delivered, not the energy of each photon.
Calculate the energy of one photon
Substitute the frequency of a single radiation component into E = hf. Frequency is measured in hertz (s⁻¹), so J s × s⁻¹ gives joules. If a source emits N identical photons, the total emitted energy is NE, but E = hf still refers to one photon.
Visible light has a range of frequencies, so its photons do not all have the same energy. Keep photon energy separate from intensity: at fixed frequency, greater intensity means more energy arriving per second because more photons arrive, not because each photon becomes more energetic.
Pause and explain
If photon frequency doubles, what happens to the energy of each photon?
Worked example
Find the energy of one photon with frequency 6.0 × 10¹⁴ Hz. Use h = 6.63 × 10⁻³⁴ J s.
Show the worked solution
- Use E = hf for one photon.
- Substitute h and f in SI units.
- Check that the result is in joules.
Answer E = (6.63 × 10⁻³⁴)(6.0 × 10¹⁴) = 3.98 × 10⁻¹⁹ J.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Calculate the energy of one photon with frequency 5.0 × 10¹⁴ Hz.
Give me a hint
Multiply frequency by Planck's constant h = 6.63 × 10⁻³⁴ J s.
Compare my reasoning
- E = hf.
- E = (6.63 × 10⁻³⁴)(5.0 × 10¹⁴).
- E = 3.315 × 10⁻¹⁹ J.
E ≈ 3.32 × 10⁻¹⁹ J per photon.
Look for these in your work
- Photon equation selected
- Substitution shown
- Energy and unit stated
A source emits two million photons per second, each with energy 4.0 × 10⁻¹⁹ J. Find the energy emitted per second.
Give me a hint
Multiply the energy per photon by the number of photons emitted each second.
Compare my reasoning
- N per second = 2.0 × 10⁶.
- Energy per second = (2.0 × 10⁶)(4.0 × 10⁻¹⁹ J).
- The result is power in joules per second.
8.0 × 10⁻¹³ J s⁻¹, or 8.0 × 10⁻¹³ W.
Look for these in your work
- Photon count rate used
- Total energy rate calculated
- Power unit identified
Two monochromatic beams have the same photon arrival rate, but beam B has twice the frequency of beam A. Compare their energy delivered per second.
Give me a hint
First compare energy per photon, then apply the equal photon arrival rates.
Compare my reasoning
- E = hf, so each B photon has twice the energy of an A photon.
- The beams have the same number of photons arriving per second.
- Therefore beam B delivers twice the energy per second.
Beam B delivers twice the energy per second.
Look for these in your work
- Frequency-energy relation applied
- Equal rates recognized
- Total energy-rate comparison stated
Common mistakes
- Using wavelength in E = hf without first finding frequency
- Treating the energy of a beam as the energy of one photon
- Assuming higher intensity at fixed frequency makes each photon more energetic
- Omitting the joule unit
What can you explain now?
In the photon model, what is one photon?
Compare with the explanation
A discrete quantum of electromagnetic energy
Radiation energy is transferred in packets; a photon is one packet.
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Make it stick
How does E = hf distinguish photon energy from the intensity of a beam?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.