Why classical physics was not enough
Classical physics (Newton's laws, Maxwell's waves) worked well for everyday objects. Around 1900 some results did not fit:
- Black-body radiation: the colours given off by hot objects could not be explained by waves alone.
- Photoelectric effect: light frees electrons only above a certain frequency, however bright it is.
- Line spectra: atoms give off only certain colours.
- Michelson–Morley experiment: the speed of light did not change with Earth's motion.
Two new theories answered these: quantum theory (for the very small) and relativity (for the very fast). Together they are modern physics. Classical physics is still correct as a special case for slow, large objects.
Quantum ideas: photons and the photoelectric effect
In 1900 Max Planck proposed that energy is given out in packets called quanta. For light, one packet is a photon with energy
E = hf = hc/λ, where h = 6.63 × 10⁻³⁴ J s.
In 1905 Einstein used photons to explain the photoelectric effect: one photon gives all its energy to one electron. If hf is less than the metal's work function φ (the energy needed to escape), no electron leaves. Otherwise
KEmax = hf − φ.
Brighter light means more photons, so more electrons, but not faster ones. A useful unit is the electronvolt: 1 eV = 1.6 × 10⁻¹⁹ J. Compton scattering (X-rays bouncing off electrons and losing energy) also showed that photons carry momentum.
Wave–particle duality and the quantum atom
Light shows interference (a wave) and the photoelectric effect (a particle). In 1924 de Broglie said matter also has a wave side: λ = h / p = h / mv. Electron diffraction proved it. In the double-slit experiment, single electrons land as dots, but many dots build up wave-like stripes. This is wave–particle duality.
Quantum mechanics describes particles with probabilities: we can predict where an electron is likely to land, not exactly where. The uncertainty principle (Heisenberg) says we cannot know both position and momentum exactly at once. In atoms, electrons have only certain energy levels; jumping between them emits or absorbs photons of exact energies, which gives line spectra.
Special relativity
Einstein's two postulates (1905):
- The laws of physics are the same in all frames moving at constant velocity.
- The speed of light in vacuum, c = 3.0 × 10⁸ m/s, is the same for all observers.
Results, using the Lorentz factor γ = 1 / √(1 − v²/c²):
- Time dilation: a moving clock runs slow. t = γt₀ (t₀ is the proper time, measured by the clock that moves with the event).
- Length contraction: L = L₀ / γ, along the direction of motion.
- Mass–energy equivalence: E = mc². A tiny mass is a huge energy; this powers the Sun and nuclear reactors.
Evidence: fast muons from cosmic rays reach the ground because their clocks run slow; atomic clocks flown on planes; GPS corrections.
Modern physics in technology and society
- Quantum: lasers, LEDs, solar cells, transistors and computer chips, electron microscopes, MRI scanners, quantum computers being built today.
- Relativity: GPS, particle accelerators, nuclear power (E = mc²), PET scans.
These bring benefits and questions: nuclear power is low-carbon but makes long-lived waste; nuclear weapons raised serious moral debates; quantum computers may one day break today's encryption.
Try it
Look at a solar garden light or calculator: cover it with red cellophane, then blue. Which colour lets it work better? Blue photons carry more energy. Then use the free-play step to predict at which frequency electrons start to escape.
Key formulas and definitions
- E = hf = hc/λ (photon energy), h = 6.63 × 10⁻³⁴ J s
- KEmax = hf − φ (photoelectric equation)
- Threshold frequency f₀ = φ / h
- 1 eV = 1.6 × 10⁻¹⁹ J
- λ = h / p = h / mv (de Broglie wavelength)
- γ = 1 / √(1 − v²/c²)
- t = γt₀ (time dilation); L = L₀/γ (length contraction)
- E = mc², c = 3.0 × 10⁸ m/s
Worked examples
1. Find the energy of a photon of frequency 6.0 × 10¹⁴ Hz, in J and eV.
E = hf = 6.63 × 10⁻³⁴ × 6.0 × 10¹⁴ = 3.98 × 10⁻¹⁹ J. In eV: 3.98 × 10⁻¹⁹ ÷ 1.6 × 10⁻¹⁹ ≈ 2.49 eV.
2. Light of photon energy 3.5 eV falls on a metal with work function 2.3 eV. Find the maximum kinetic energy of the electrons.
KEmax = hf − φ = 3.5 − 2.3 = 1.2 eV.
3. A metal has φ = 2.3 eV. Will red light of 4.5 × 10¹⁴ Hz free electrons?
E = 6.63 × 10⁻³⁴ × 4.5 × 10¹⁴ = 2.98 × 10⁻¹⁹ J ≈ 1.86 eV. This is less than 2.3 eV, so no electrons, however bright.
4. Find the de Broglie wavelength of an electron (m = 9.1 × 10⁻³¹ kg) moving at 1.0 × 10⁶ m/s.
p = mv = 9.1 × 10⁻²⁵ kg m/s. λ = h/p = 6.63 × 10⁻³⁴ ÷ 9.1 × 10⁻²⁵ ≈ 7.3 × 10⁻¹⁰ m (about the size of an atom, so electrons diffract from crystals).
5. A spaceship moves at 0.8c. Find γ. If 1 hour passes on the ship, how long passes on Earth?
γ = 1/√(1 − 0.64) = 1/√0.36 = 1/0.6 ≈ 1.67. Earth time t = γt₀ = 1.67 × 1 h ≈ 1.67 h (100 minutes).
6. How much energy is in 1 g of mass (E = mc²)?
E = 0.001 × (3.0 × 10⁸)² = 0.001 × 9 × 10¹⁶ = 9 × 10¹³ J, about the energy of a large power station running for a day.
Common mistakes
- Thinking brighter light gives faster photoelectrons. Brightness changes the number of electrons; frequency changes their energy.
- Using the wrong time in t = γt₀: t₀ is the proper time on the moving clock, and t is always longer.
- Thinking relativity only matters for spaceships. GPS and particle accelerators need it every day.
- Saying light is "either" a wave "or" a particle. It shows both behaviours, depending on the experiment.