Modern Physics — Relativity & Quantum
CollegeSpecial relativity and quantum mechanics
Einstein's special relativity (1905) rests on two postulates: the laws of physics are the same in all inertial frames, and the speed of light c is the same for every observer. Startling consequences follow, governed by the Lorentz factor γ = 1/√(1 − v²/c²):
L = L₀/γ (length contraction)
E = mc² (mass–energy equivalence)
Quantum mechanics governs the very small. Energy is quantised: light comes in photons of energy E = hf (h = 6.63×10⁻³⁴ J·s). Matter has a wave nature, wavelength λ = h/p (de Broglie). Heisenberg's uncertainty principle forbids exact simultaneous knowledge of position and momentum:
Why reality is stranger than intuition
Relativity says time is not absolute. A fast-moving clock runs slow relative to you — not an illusion but a real, measured effect. GPS satellites must correct for it: their clocks tick faster (weaker gravity, general relativity) and slower (their speed) by tens of microseconds a day; uncorrected, GPS would drift ~10 km daily. E = mc² means mass is concentrated energy — the source of the Sun's power and nuclear energy.
Quantum mechanics reveals wave–particle duality: electrons and photons behave as waves (interference in double slits) and particles (discrete photon impacts). The photoelectric effect — light ejecting electrons only above a threshold frequency, regardless of intensity — proved light is quantised and won Einstein the Nobel Prize. At quantum scale we can predict only probabilities, not certainties; nature is fundamentally statistical.
Worked examples — time dilation and the photoelectric effect
Part A — time dilation. A muon travels at 0.99c. Its rest lifetime is 2.2 μs. How long does it live in the lab frame?
- Lorentz factor:
γ = 1/√(1 − 0.99²) = 1/√(1 − 0.9801) = 1/√0.0199 = 7.09. - Dilated lifetime:
Δt = γΔt₀ = 7.09 × 2.2 μs = 15.6 μs. - This is why cosmic-ray muons reach the ground despite a "too-short" bare lifetime — a real, daily confirmation of relativity.
Part B — photoelectric effect. A metal has work function φ = 2.3 eV. Light of wavelength 400 nm shines on it. Find the maximum kinetic energy of ejected electrons.
- Photon energy:
E = hc/λ = (6.63×10⁻³⁴ × 3.0×10⁸)/(400×10⁻⁹) = 4.97×10⁻¹⁹ J. - Convert to eV:
4.97×10⁻¹⁹ / 1.60×10⁻¹⁹ = 3.11 eV. - Einstein's equation:
KEmax = hf − φ = 3.11 − 2.3 = 0.81 eV. - Since the photon energy (3.11 eV) exceeds the work function, electrons are ejected. Below
φ, none would be — no matter how bright.
How much energy is stored in 1 gram of matter, per E = mc²?
E = mc² = 0.001 × (3.0×10⁸)² = 0.001 × 9.0×10¹⁶ = 9.0×10¹³ J — about 21 kilotonnes of TNT, or the energy to power a city for hours. This is the colossal energy density of mass, tapped in nuclear reactions.
Why did the photoelectric effect require the photon idea?
Classical waves predict brighter light ejects electrons with more energy. Experiment shows electron energy depends on frequency, not intensity, with a sharp threshold. Einstein explained this with light quanta of energy hf: one photon ejects one electron, and below the threshold frequency no single photon has enough energy — intensity is irrelevant.
What does the uncertainty principle really claim?
Not merely a measurement limitation — it is fundamental. A particle cannot simultaneously possess a precise position and precise momentum: Δx·Δp ≥ ℏ/2. Confine a particle tightly (small Δx) and its momentum becomes correspondingly uncertain (large Δp). This underlies atomic stability and quantum tunnelling. Chapter 21 derives this rigorously from operator algebra.
Twin A travels near light-speed and returns; twin B stays home. Who is older?
Twin B (who stayed) is older. The travelling twin's clock ran slow due to time dilation, so less time passed for them. The situation isn't symmetric because the travelling twin accelerated to turn around, breaking the equivalence — resolving the famous "twin paradox". Chapter 18 develops the proper-time formalism behind this.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1A clock moves at 0.60c. Find its time dilation factor gamma.
- gamma = 1/sqrt(1 - v^2/c^2)
- = 1/sqrt(1 - 0.36)
- = 1/sqrt(0.64) = 1/0.8
Example 2Light of frequency 6.0e14 Hz hits a metal of work function 2.0 eV. Find the photon energy in eV. (h=6.63e-34, 1 eV=1.6e-19 J)
- E = h*f = 6.63e-34 * 6.0e14 = 3.98e-19 J
- In eV: 3.98e-19 / 1.6e-19
Now you try
Work each one out first, then tap to reveal the worked answer.