Modern Physics: Relativity and Quantum Mechanics

Grade 11 · Science · Study Guide

Explore Einstein's special relativity, time dilation, length contraction, and the foundational principles of quantum mechanics.

Modern physics revolutionized our understanding of nature in the early 20th century. Einstein's Special Theory of Relativity (1905) rests on two postulates: the laws of physics are the same in all inertial reference frames, and the speed of light in a vacuum (c ≈ 3×10⁸ m/s) is the same for all observers regardless of the source's motion.

These postulates lead to remarkable consequences. Time dilation means a moving clock ticks more slowly: Δt = γΔt0, where γ = 1/√(1 - v²/c²) is the Lorentz factor and Δt0 is the proper time (measured in the rest frame). Length contraction means a moving object appears shorter along the direction of motion: L = L0/γ. Mass-energy equivalence is captured by E = mc², showing mass is a form of energy.

Quantum mechanics describes nature at atomic and subatomic scales. Max Planck proposed that energy is quantized in discrete packets called quanta: E = hf, where h is Planck's constant (6.626×10⁻³⁴ J·s) and f is frequency. The photoelectric effect, explained by Einstein, showed light behaves as particles (photons), each carrying energy hf.

De Broglie proposed wave-particle duality: matter also has wave properties with wavelength λ = h/p. Heisenberg's Uncertainty Principle states that the position and momentum of a particle cannot both be known precisely simultaneously: ΔxΔp ≥ h/(4π). Bohr's model of the hydrogen atom quantized orbital energy levels, predicting the observed emission spectrum.

Key terms

Lorentz Factor (γ): γ = 1/√(1 - v²/c²); a factor greater than 1 for moving objects that scales relativistic time dilation and length contraction.
Time Dilation: Moving clocks run slower: Δt = γΔt0; a consequence of special relativity where time passes more slowly for a moving observer.
Photoelectric Effect: Emission of electrons from a material when light above a threshold frequency strikes it; proves light's particle nature.
Planck's Constant (h): h = 6.626×10⁻³⁴ J·s; the fundamental quantum of action; relates photon energy to frequency via E = hf.
Wave-Particle Duality: Both light and matter exhibit both wave-like and particle-like properties depending on how they are observed or measured.
Heisenberg's Uncertainty Principle: ΔxΔp ≥ h/(4π); the position and momentum of a quantum particle cannot both be measured with arbitrary precision simultaneously.

Flashcards

Special Relativity Postulates: 1) Laws of physics are the same in all inertial frames. 2) Speed of light c is constant for all observers.
Lorentz Factor: γ = 1/√(1 - v²/c²); always ≥ 1; approaches infinity as v → c.
Time Dilation Formula: Δt = γΔt0; moving clocks run slow. Δt0 is proper time (rest frame), Δt is dilated time (observer frame).
Length Contraction Formula: L = L0/γ; moving objects appear shortened along direction of motion. L0 is the proper length.
Mass-Energy Equivalence: E = mc²; mass and energy are interchangeable; c² ≈ 9×10¹⁶ J/kg converts tiny masses to enormous energies.
Photoelectric Effect: Light of frequency f above threshold ejects electrons with max KE = hf - φ, where φ is the work function.
de Broglie Wavelength: λ = h/p = h/(mv); all matter has an associated wavelength; significant only for subatomic particles.
Heisenberg Uncertainty Principle: ΔxΔp ≥ ℏ/2 (or h/4π); fundamental limit on simultaneous precision of position x and momentum p.

Questions

A spaceship travels at 0.6c relative to Earth. Its proper length is 100 m. What is its observed length from Earth?
1. 100 m
2. 80 m
3. 60 m
4. 120 m
Show answer
Answer: 80 m
γ = 1/√(1 - 0.36) = 1/√0.64 = 1/0.8 = 1.25. L = L0/γ = 100/1.25 = 80 m.
In the photoelectric effect, increasing the intensity of light (without changing frequency) increases the kinetic energy of emitted electrons.
Show answer
Answer: False
Increasing intensity means more photons per second, causing more electrons to be ejected (greater current), but each photon still has energy hf. Only increasing frequency increases the KE of individual electrons.
The energy of a photon is given by E = hf, where h is Planck's constant and f is the photon's _____.
Show answer
Answer: frequency
E = hf relates photon energy to frequency f. Higher frequency (shorter wavelength) means higher energy photons.
An observer on Earth watches a clock on a spaceship moving at 0.8c. If the spaceship clock shows 10 seconds passing, how many seconds pass on Earth?
1. 6 s
2. 10 s
3. 16.7 s
4. 12.5 s
Show answer
Answer: 16.7 s
γ = 1/√(1 - 0.64) = 1/√0.36 = 1/0.6 ≈ 1.667. Δt = γΔt0 = 1.667 × 10 ≈ 16.7 s. The Earth observer sees more time pass.
De Broglie proposed that all moving particles have an associated wavelength inversely proportional to their momentum.
Show answer
Answer: True
De Broglie's hypothesis: λ = h/p = h/(mv). Since λ is inversely proportional to momentum p, higher momentum particles have shorter wavelengths.

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