Modern Physics Study Guide

Level High SchoolReview time ~12 hours

Modern physics rewrites common sense: time stretches, mass becomes energy, light arrives in packets, and particles refuse precise addresses. The mathematics is small — a few famous equations — but the concepts demand careful thinking. This guide keeps both straight.

Key formulas

Kinetic Energy Formula — KE = ½mv²

Kinetic energy equals one-half times mass times velocity squared.

Mass-Energy Equivalence — E = mc²

Energy equals mass times the speed of light squared.

Wave Equation — v = fλ

Wave speed equals frequency times wavelength.

Work Formula — W = Fd

Work equals force times distance moved in the direction of the force.

Topic checklist

Special relativity basics

State Einstein's two postulates of special relativity.
Explain why the speed of light is the same for all observers.
Distinguish inertial from non-inertial frames.

Time dilation and length contraction

Use $\Delta t = \gamma\Delta t_0$ with $\gamma = 1/\sqrt{1-v^2/c^2}$ .
Identify the proper time (measured in the moving frame).
Explain the twin paradox resolution: acceleration breaks the symmetry.

Mass-energy equivalence

Apply $E = mc^2$ to mass-energy conversions.
Convert fluently between kilograms, joules, and electronvolts.
Explain why chemical reactions show negligible mass change.

Photoelectric effect

Describe the photoelectric effect: photons eject electrons above a threshold frequency.
Use $KE_{max} = hf - \Phi$ where $\Phi$ is the work function.
Explain why this experiment forced the photon concept.

Wave-particle duality

State wave-particle duality: everything has both natures.
Use de Broglie's $\lambda = h/p$ for matter waves.
Explain the double-slit result for single particles.

Uncertainty and nuclear decay

State $\Delta x\,\Delta p \ge \hbar/2$ in words.
Explain that uncertainty is fundamental, not instrument error.
Describe half-life and the randomness of individual decays.

Common mistakes

Mistake: Treating the speed of light as frame-dependent.

Fix:

Relativity says $c$ is invariant; everything else (time, length, simultaneity) flexes to protect that. Start every relativity problem by asking which frame measures proper time or proper length.

Mistake: Applying time dilation and length contraction to the wrong frame.

Fix:

Time dilation stretches the moving clock's ticks as seen by you; length contraction shortens the moving object's length as seen by you. Each observer sees the other's clocks slow — the symmetry breaks only with acceleration.

Mistake: Thinking mass disappears in nuclear reactions.

Fix:

$E = mc^2$ converts mass to energy and back; in nuclear reactions the 'missing' mass leaves as kinetic energy and radiation. Track total energy (mass plus kinetic), and the books balance.

Mistake: Believing brighter light always ejects faster electrons.

Fix:

The work function $\Phi$ is an energy threshold per photon, not per beam. Below the threshold frequency no electrons emerge no matter how bright the light — brightness adds photons, not energy per photon.

Mistake: Calling the uncertainty principle a measurement limitation.

Fix:

Uncertainty is built into quantum mechanics: a particle simply does not have a precise position and momentum simultaneously. Better instruments cannot beat $\hbar/2$ .

Exam tips

Write down which frame is which before using gamma — most errors are frame mix-ups.
Convert eV to joules (1 eV = 1.602e-19 J) before mixing with SI formulas.
For photoelectric problems, check the threshold frequency first — it can short-circuit the whole question.
Half-life problems: count half-lives as powers of two, then multiply by the half-life.