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.