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Modern Physics — Formula Cheat Sheet
Every key formula for Modern Physics, with what it means and where it comes from.
| Formula | What it means | Topic |
|---|---|---|
| |psi> = alpha |0> + beta |1> | A qubit state is a superposition of 0 and 1, with amplitudes alpha and beta. | Quantum Computing Basics: Qubits and Superposition |
| |alpha|^2 + |beta|^2 = 1 | The probabilities of measuring 0 and 1 must add up to one. | Quantum Computing Basics: Qubits and Superposition |
| 2^n states | n qubits can be in a superposition of 2^n basis states at once. | Quantum Computing Basics: Qubits and Superposition |
| E = h f | Energy of a quantum of light, with Planck constant h = 6.63 x 10^-34 J s. | Quantum Mechanics: An Introduction |
| lambda = h / p | de Broglie wavelength of a particle with momentum p. | Quantum Mechanics: An Introduction |
| delta x delta p >= h / (4 pi) | Heisenberg uncertainty principle: position and momentum cannot both be known exactly. | Quantum Mechanics: An Introduction |
| N = N0 (1/2)^(t / T_half) | Nuclei remaining after time t, with half-life T_half. | Radioactivity and Half-Life |
| A = lambda N | Activity equals decay constant times number of undecayed nuclei. | Radioactivity and Half-Life |
| lambda = ln 2 / T_half | Decay constant from half-life, with ln 2 = 0.693. | Radioactivity and Half-Life |
| delta t = gamma delta t0 | Time dilation: a moving clock runs slow, with gamma = 1 / sqrt(1 - v^2/c^2). | Relativity: Time Dilation and Length Contraction |
| L = L0 / gamma | Length contraction: a moving object is shorter along its direction of motion. | Relativity: Time Dilation and Length Contraction |
| E = m c^2 | Mass-energy equivalence: rest energy equals mass times the speed of light squared. | Relativity: Time Dilation and Length Contraction |
| gamma = 1 / sqrt(1 - v^2 / c^2) | Lorentz factor, which grows as speed approaches c. | Relativity: Time Dilation and Length Contraction |
| R = 0 for T < Tc | Electrical resistance is zero below the critical temperature Tc. | Superconductivity Explained |
| B = 0 inside | Meissner effect: a superconductor expels magnetic field from its interior. | Superconductivity Explained |
| E_n = -13.6 / n^2 eV | Energy of level n in hydrogen, with n = 1, 2, 3, ... | The Bohr Model of the Atom |
| r_n = n^2 a0 | Radius of orbit n, with Bohr radius a0 = 5.29 x 10^-11 m. | The Bohr Model of the Atom |
| h f = E_upper - E_lower | Photon energy equals the difference between two energy levels. | The Bohr Model of the Atom |
| KE_max = h f - phi | Maximum kinetic energy of an emitted electron, with photon energy hf and work function phi. | The Photoelectric Effect |
| h f0 = phi | Threshold condition: the threshold frequency f0 is where photon energy just equals the work function. | The Photoelectric Effect |
| E = h f | Energy of one photon. | The Photoelectric Effect |
| P = σ × A × T⁴ | Radiated power grows with the fourth power of absolute temperature, so a small temperature rise boosts the glow sharply. | Black Body Radiation Calculator |
| E_n = −13.6 ÷ n² eV | Hydrogen's electron can only sit in fixed energy levels, and the level energy is negative because the electron is bound. | Bohr Energy Level Calculator |
| N = N₀ × (½)^(t ÷ T½) | After each half-life the undecayed amount halves, so the remaining quantity falls exponentially with time. | Half-Life Calculator |
| E = Δm × c² | A nucleus weighs slightly less than its separate parts, and that missing mass equals the binding energy via E equals mc squared. | Nuclear Binding Energy Calculator |
| KE_max = h × f − φ | An ejected electron's energy is the photon energy minus the metal's work function, and it can be zero or negative below threshold. | Photoelectric Effect Calculator |
| λ = h ÷ p | Every moving particle has a wavelength set by Planck's constant divided by its momentum, mass times velocity. | de Broglie Wavelength Calculator |