Modern Physics
E=mc², photoelectric. Every formula below opens into a live, hands-on simulation.
Mass-Energy Equivalence
Mass is a highly concentrated form of energy; the speed of light squared is the conversion factor.
Lorentz Factor
As speed approaches c, time stretches and lengths contract by the factor γ.
Photoelectric Effect
Light comes in quanta of energy hν; only photons above the work-function threshold can free electrons.
De Broglie Wavelength
Every particle has a wavelength inversely proportional to its momentum — heavy/fast things have wavelengths so small they're unobservable.
Heisenberg Uncertainty Principle
Position and momentum cannot both be sharply defined; nature enforces a fundamental fuzziness at small scales.
Schrödinger Equation (Time-Independent)
The total energy operator acting on the wavefunction returns the energy times the same wavefunction — an eigenvalue problem for reality.
Bohr Energy Levels (Hydrogen)
Electrons in hydrogen are stuck on a ladder of negative energies; the gaps determine the colors of light atoms emit.
Radioactive Decay Law
Each nucleus has a fixed probability per unit time of decaying, producing a smooth exponential decline in the population.
Compton Scattering
Photons carry momentum p = h/λ. When one collides with a free electron, conservation of energy + momentum forces the photon to give up energy — its wavelength grows by an amount that depends only on the scattering angle.
Bohr Radius
A balance between two demands: Coulomb attraction wants the electron as close to the proton as possible, but the uncertainty principle penalizes localization (smaller box → bigger momentum → bigger kinetic energy). The minimum-energy compromise sits at a₀.
Rydberg Formula
Each integer n labels an allowed electron energy level (the staircase). A photon's wavelength encodes the energy difference between two steps. Two integers → all hydrogen spectral lines.
Planck Radiation Law
Light energy is quantized in packets of size hν. At low frequency, packets are cheap → many emitted (Rayleigh-Jeans). At high frequency, each packet costs more than kT → exponentially suppressed. The peak balance gives the body's color.
Wien's Displacement Law
Hotter → more energetic photons → shorter peak wavelength. The exact peak comes from differentiating Planck's law and solving a transcendental equation; the answer is a universal product b.
Stefan-Boltzmann Law
Integrating the Planck spectrum over all wavelengths gives the total emitted power. Two factors of T from the peak shift (Wien) and two more from the bandwidth growth combine to a T⁴ scaling.
Pair Production Threshold
Mass-energy equivalence says creating two particles of mass m_e demands at least 2m_e·c² of energy. But a lone photon can't do it — momentum conservation forbids it. A nearby nucleus (or another photon) absorbs the recoil and unlocks the process.
Time Dilation
The speed of light is the same for all observers. To keep that fixed when one observer moves relative to another, time itself must stretch — moving clocks run slow.
Length Contraction
Just as moving clocks dilate, moving rulers contract. Both follow from c being invariant: lengths along motion direction shrink by 1/γ. Lengths perpendicular to motion are unchanged.
Planck's Blackbody Radiation Law
Heat any object and it glows — dull red, then orange, then white-hot. Classical physics predicted the glow should carry infinite energy at short wavelengths (the 'ultraviolet catastrophe'). Planck fixed it by a desperate guess: light energy comes in discrete lumps hf, not a continuum. That single assumption bends the spectrum back down at short wavelengths and matches every glowing object perfectly. It was the first crack in classical physics and the birth of the quantum.
Work Function & Stopping Potential
Shine light on a metal and electrons pop out — but only if the light's color is bluer than a threshold, no matter how bright. Einstein explained it: light arrives as packets of energy hf. One packet kicks out one electron, but the electron must first pay an 'exit toll' φ to escape the metal. Whatever energy is left over becomes the electron's kinetic energy. Brightness adds more packets (more electrons) but never more energy per electron — that only comes from higher frequency.
Wave-Particle Duality (de Broglie)
If light — long thought a wave — can act like particles (photons), de Broglie asked the reverse: can matter act like waves? He assigned every particle a wavelength λ = h/p. For a baseball this wavelength is absurdly tiny and invisible; for an electron it's about the size of an atom, so electrons diffract and interfere just like light. Everything is both, but Planck's constant is so small that the wave side only shows up for the very light and very slow.
Rutherford Scattering
Rutherford fired alpha particles at gold foil expecting them to pass almost straight through the 'plum-pudding' atom. Instead a few bounced almost straight back — 'as if you fired a 15-inch shell at tissue paper and it came back'. The only way to deflect a heavy fast alpha so sharply is a tiny, dense, positive core: the nucleus. The 1/sin⁴(θ/2) law is the fingerprint of a point Coulomb charge, and its success pinned down the nuclear atom.
Band Theory of Solids
Bring one atom's sharp energy level close to another's and they split in two; bring 10²³ atoms together in a crystal and the levels smear into continuous bands separated by forbidden gaps. Whether a material conducts depends on where the electrons stop filling: land in the middle of a band and it's a metal; fill a band exactly to a gap and it's an insulator or, if the gap is small, a semiconductor. The band gap E_g is the single number that decides.
Drude Free-Electron Model
Drude pictured a metal as a gas of free electrons rattling around fixed positive ions, like pinballs. Apply a voltage and the electrons accelerate, but keep crashing into ions every time τ; between crashes they pick up a small average drift velocity. Balancing the electric push against the collision drag gives Ohm's law and a formula for conductivity. It's crude — it ignores quantum statistics — yet it nails the form of σ and even predicts the Hall effect.
Quantization of Electric Charge
Charge doesn't come in arbitrary amounts — it comes in whole-number multiples of one indivisible unit e. Millikan proved it by suspending tiny charged oil droplets between charged plates: turn up the voltage until an electron's electric pull exactly balances gravity. Every drop's charge came out as an integer times the same tiny number, 1.6×10⁻¹⁹ C. You never find half an electron's worth. Charge is granular, like money in indivisible pennies.
Fine-Structure Constant
α is the single dimensionless number that measures how strongly light and electrons couple — the strength of electromagnetism itself. Because it's a pure number (~1/137) with no units, every civilization in the universe would measure the same value. It sets the fine splitting of spectral lines, the speed of an electron in the first Bohr orbit (αc), and why atoms are the size they are. Feynman called it 'one of the greatest damn mysteries of physics.'
Balmer Series
Hydrogen's visible glow is four sharp lines — red, cyan, blue, violet. In 1885 a Swiss schoolteacher, Balmer, found a simple formula fitting their wavelengths using integers, with no physical theory at all. Every line is an electron falling to the n=2 level from a higher orbit; the bigger the drop, the bluer the light. Bohr later explained why the integers appear. The Balmer series is the training-wheels case that cracked open atomic structure.
Bohr Magneton
An electron orbiting or spinning is a tiny current loop, and every current loop is a magnet. The Bohr magneton is the natural unit of that magnetism — the magnetic moment of an electron with one quantum of angular momentum. Atomic magnets come in multiples of this unit, so μ_B is to magnetism what e is to charge: the fundamental brick. It sets the scale of the Zeeman effect, electron spin resonance, and the strength of ferromagnets.