§ 22.1 Energy and momentum of a photon
Key ideas
- Electromagnetic radiation comes in photons, each a quantum of energy E = hf = hc/λ.
- The electronvolt is the energy gained by an electron moving through 1 V: 1 eV = 1.6 × 10⁻¹⁹ J.
- A photon carries momentum p = E/c even though it has no mass.
Equations
E = h f = h c / λh = 6.63 × 10⁻³⁴ J sJ
p = E / cmomentum of a photonkg m s⁻¹
Watch out: energy and frequency are proportional, but energy and wavelength are inversely proportional. Short-wavelength photons (ultraviolet, X-rays) carry the most energy, not the least.
§ 22.2 Photoelectric effect
Key ideas
- Light above a threshold frequency ejects electrons from a metal instantly; below it, none are emitted however bright the light.
- The work function Φ is the minimum energy to free an electron. Einstein's equation is energy conservation for one photon and one electron.
- The maximum kinetic energy depends on frequency, not intensity; brighter light of the same colour ejects more electrons, not faster ones.
Equations
h f = Φ + ½ m v²_maxEinstein's photoelectric equationJ
f₀ = Φ / hthreshold frequencyHz
Fig. 1 · The photoelectric graph: KE_max rises linearly with frequency above the threshold f₀, with gradient h (Planck's constant) and an intercept of −Φ.
Watch out: increasing the intensity raises the number of photoelectrons (and the current), but never their maximum kinetic energy. Only a higher frequency gives faster electrons.
§ 22.3 Wave-particle duality
Key ideas
- Light shows particle behaviour (the photoelectric effect) and wave behaviour (interference and diffraction).
- Electron diffraction shows that particles also have a wave nature, with a de Broglie wavelength set by their momentum.
Equations
λ = h / p = h / (m v)de Broglie wavelength of a particlem
Watch out: faster electrons have a shorter wavelength (λ = h/mv), so they diffract less. Speeding the electrons up shrinks the diffraction rings, it does not widen them.
§ 22.4 Energy levels in atoms and line spectra
Key ideas
- Electrons in an isolated atom occupy discrete energy levels, labelled with negative energies (bound states).
- A photon is emitted or absorbed only when its energy exactly matches a gap between levels, giving sharp line spectra.
- Emission lines (bright) and absorption lines (dark) appear at the same wavelengths for a given element.
Equations
h f = E₁ − E₂photon energy equals the level gapJ
Fig. 2 · Energy levels: an electron dropping from a higher level to a lower one emits a photon whose energy is exactly the gap, hf = E₃ − E₁.
Watch out: energy levels are negative and get closer together towards the top. The gap E₁ − E₂ should be taken as a positive photon energy; mind the signs when subtracting two negative levels.