Quantum Physics

This guide covers Topic 22, Quantum physics, in full — subtopics 22.1 Energy and momentum of a photon, 22.2 Photoelectric effect, 22.3 Wave-particle duality and 22.4 Energy levels in atoms and line spectra — from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is A Level content and the final electricity-and-modern-physics topic before nuclear physics.

Before studying this

This resource assumes electromagnetic waves from Waves: Progressive Waves, the Doppler Effect and Polarisation, and atomic structure from Particle Physics: Atoms, Nuclei and Fundamental Particles.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — A Level, Topic 22

22.1 Energy and momentum of a photon — understanding that electromagnetic radiation has a particulate nature, and that a photon is a quantum of electromagnetic energy; recalling and using E = hf; recalling and using the electron-volt as a unit of energy; recalling and using p = E/c and p = h/λ for the momentum of a photon.

22.2 Photoelectric effect — understanding that photoelectrons may be emitted from a metal surface when it is illuminated by electromagnetic radiation; understanding and using the terms threshold frequency and threshold wavelength; explaining photoelectric emission in terms of photon energy and work function energy; recalling and using hf = Φ + Eₖ(max); explaining why the maximum photoelectric energy is independent of intensity, whereas the photoelectric current is proportional to intensity, for radiation of a particular frequency.

22.3 Wave-particle duality — understanding that the photoelectric effect provides evidence for a particulate nature of electromagnetic radiation while phenomena such as interference and diffraction provide evidence for a wave nature, and hence that light has a dual electromagnetic wave-particle nature; understanding that electrons possess wave properties, having originally been considered to be particles; recalling and using the relation for the de Broglie wavelength λ = h/p.

22.4 Energy levels in atoms and line spectra — understanding that there are discrete electron energy levels in isolated atoms; recalling and using hf = E₁ − E₂; understanding that the emission and absorption line spectra provide evidence for the existence of discrete electron energy levels in isolated atoms.

Photon energy

Electromagnetic radiation is not only wave-like: it also transfers energy in discrete packets called photons. The energy of a single photon is:

E = hf

where h is the Planck constant. Photon energy can also be expressed in electron-volts (eV), a convenient unit at the atomic scale, where 1 eV is the energy gained by an electron accelerated through a potential difference of 1 V.

Photon momentum

Although a photon has no rest mass, it still carries momentum, related to its energy by:

p = E/c

where c is the speed of light. Combining this with E = hf and c = fλ gives an equivalent form in terms of wavelength:

p = h/λ

Worked example. A photon of wavelength 500 nm has momentum:

p = h/λ = (6.63 × 10⁻³⁴) / (500 × 10⁻⁹) = 1.33 × 10⁻²⁷ kg m s⁻¹

This is the same expression, λ = h/p, that reappears rearranged in the de Broglie relation for matter waves below — photon momentum is the historical starting point for that idea.

The photoelectric effect

When electromagnetic radiation of sufficiently high frequency shines on a metal surface, it can eject electrons — the photoelectric effect. Below a certain threshold frequency (or above a corresponding threshold wavelength), no electrons are emitted, no matter how intense the radiation. This is explained by treating light as photons: each photon interacts with a single electron, transferring all its energy at once. If this energy is at least the metal’s work function Φ (the minimum energy needed to free an electron from the surface), an electron is emitted; the remaining energy becomes its kinetic energy:

hf = Φ + Eₖ(max)

Because each photon-electron interaction is independent of every other, increasing the intensity of radiation (more photons per second, same frequency) increases the rate of electron emission (photoelectric current) but does not increase the maximum energy of any individual emitted electron — that depends only on the frequency of the radiation, through the equation above.

Worked example. A metal has a work function of 2.0 eV. Light of frequency 8.0 × 10¹⁴ Hz (E = hf = 6.63 × 10⁻³⁴ × 8.0 × 10¹⁴ ≈ 5.30 × 10⁻¹⁹ J ≈ 3.31 eV) shines on it. Maximum kinetic energy of emitted electrons:

Eₖ(max) = hf − Φ = 3.31 − 2.0 = 1.31 eV

Wave-particle duality

The photoelectric effect is strong evidence that light behaves as particles (photons); interference and diffraction (from Topic 8, Superposition) are equally strong evidence that light behaves as a wave. Both are true — this is wave-particle duality. Remarkably, the same duality applies in reverse to particles traditionally thought of as purely particulate, such as electrons: they too show wave-like behaviour (diffraction), described by the de Broglie wavelength:

λ = h/p

where p is the particle’s momentum.

Energy levels and line spectra

Electrons in an isolated atom can only occupy specific, discrete energy levels, not a continuous range. When an electron transitions between two levels, it emits or absorbs a single photon whose energy exactly matches the energy difference between the levels:

hf = E₁ − E₂

This explains line spectra: because only certain specific energy differences are possible in a given atom, only certain specific photon frequencies (and hence wavelengths) are emitted or absorbed, appearing as discrete lines rather than a continuous spread — direct experimental evidence for the existence of discrete atomic energy levels.

Common mistakes

• Assuming increasing light intensity increases the maximum kinetic energy of photoelectrons — only increasing frequency does this; intensity only affects the rate (current) of emission.
• Forgetting that no electrons are emitted at all below the threshold frequency, regardless of intensity — this is the key evidence for the particulate (photon) model, inconsistent with a purely wave-based description.
• Mixing up which equation applies where — hf = Φ + Eₖ(max) is for the photoelectric effect; hf = E₁ − E₂ is for atomic energy level transitions. They describe different physical situations.
• Treating wave-particle duality as “sometimes a wave, sometimes a particle” as an either/or — both aspects are simultaneously true properties of quantum objects, evidenced by different experiments.

Quick revision checklist

• E = hf for photon energy, and using the electron-volt
• p = E/c = h/λ for photon momentum
• hf = Φ + Eₖ(max) for the photoelectric effect, and why intensity affects current but not maximum kinetic energy
• λ = h/p, the de Broglie wavelength, showing particles have wave properties
• hf = E₁ − E₂, discrete energy levels, and line spectra as evidence for them

Related resources

• Alternating Currents — the previous A Level topic
• Nuclear Physics — the next A Level topic
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.