What is modern physics?
Classical mechanics and electromagnetism work beautifully until either speeds approach the speed of light or sizes approach atomic scales; modern physics is the regime beyond. It splits into three threads: special relativity (high speeds, where c is the speed limit and mass, length, and time depend on the observer), quantum mechanics (small scales, where energy is quantized, particles have wave-like properties, and observation itself affects outcome), and nuclear and particle physics (the binding and decay of atomic nuclei, where mass itself is a reservoir of energy). The six calculators on this panel cover the equations that fall out of the three threads — Lorentz contraction and dilation, de Broglie wavelength, the photoelectric effect, E = mc², radioactive decay, and the Bohr model of hydrogen — plus the universal constants c, h, and eV that run through all of them.
Two universal constants do almost everything here: c = 299,792,458 m/s for relativity (the speed of light is the same in every direction for every observer), and h = 6.626×10−34 J·s for quantum (every interaction exchanges energy in chunks of size hf). Add the conversion 1 eV = 1.602×10−19 J for atomic-scale work, and most of the panel's numbers fit in your head.
Special relativity — c is the limit
Einstein's 1905 postulate: the speed of light c is the same in every inertial frame, and physical laws are the same in every inertial frame. The consequence for time and length is the Lorentz factor γ = 1/√(1 − v²/c²), which inflates time (Δt = γ·T0) and shrinks length (L = L0/γ) in the moving frame. The default sample (v = 0.8 c, T0 = 1 s, L0 = 10 m) gives γ ≈ 1.667, so a clock on a fast-moving ship ticks out 1 s of proper time as 1.667 s pass for the stationary observer, and a 10 m ship appears to be 6 m long. At v = 0.5 c, γ is just 1.155 — barely noticeable; at v = 0.99 c, γ is 7.09 and relativistic effects are dramatic. The GPS network has to correct for satellite-clock time dilation at orbital speeds (~45 μs/day slower) and gravitational blueshift (~45 μs/day faster), and the two effects nearly cancel but not exactly — uncorrected they would put a GPS fix off by about 10 km after one day.
de Broglie wavelength — matter has waves
Light is a wave and also a particle; by symmetry, every particle with momentum p is also a wave with wavelength λ = h/p = h/(mv). The default sample (an electron at v = 106 m/s) gives λ ≈ 0.727 nm, on the X-ray/soft-X-ray border. This is the resolution limit for an electron microscope — and why electron microscopy beats light microscopy by orders of magnitude. For a baseball at 40 m/s (160 km/h), λ ≈ 10−34 m, many orders smaller than a proton, which is why baseballs do not diffract through fences. The wave-particle duality is universal; only the constant h keeps everyday matter from looking like a quantum object.
Photoelectric effect — quantization proved
Light shone on a metal ejects electrons, but only if the photon energy exceeds the metal's work function φ: KE = hf − φ, where f is the light frequency and h is Planck's constant. The default sample (a copper-like φ = 4.3 eV surface, λ = 200 nm ultraviolet) gives a photon energy of hf = 1240/200 = 6.2 eV and electron kinetic energy of 1.9 eV, with an electron velocity of about 8.16×105 m/s. Below the threshold frequency (φ/h in Hz, or 1240/φ in nm-eV — 288 nm for copper) no electrons emerge regardless of intensity. The effect was the experiment that settled the wave-particle debate in 1905, because classical physics predicts electron energy should depend on intensity, not frequency.
E = mc² — mass is energy
The mass-energy equivalence E = mc² ties mass to energy with a huge proportionality constant (c² = 8.988×1016 J/kg). One kilogram of mass holds 8.988×1016 J, the energy released by a 21.5-megaton thermonuclear weapon — or, equivalently, the daily energy use of the entire world economy (~5×1020 J) would require only about 5,500 kg of mass annihilation. In practice the conversion is never complete: nuclear fission releases about 0.1% of mass-energy, nuclear fusion about 0.7%, and matter-antimatter annihilation the full 100%. The mass defect (the difference between the sum of nucleon masses and the actual nucleus mass) is the binding energy released when those nuclei form, which is why nuclear binding energy tables sit next to mass tables in every nuclear data handbook.
Half-life — the probabilistic clock
Every unstable nucleus has a fixed probability of decaying per unit time, independent of its age or its chemical environment. The result is exponential decay: N(t) = N0·(1/2)t/T½. The default sample (N0 = 1000 atoms, T½ = 10 s, t = 25 s) gives N ≈ 176.78 survivors (about 17.7%) and 823.22 decayed (82.3%), after 2.5 half-lives. The mean lifetime is τ = T½/ln(2) ≈ 1.443·T½ (the average atom survives longer than the half-life because some live much longer than the median). The decay constant λ = ln(2)/T½ is the per-unit-time probability; the activity A = λ·N is decays per second, the unit of which is the becquerel (1 Bq = 1 decay/s). Carbon-14 dating uses this exact relation with T½ = 5,730 years to date organic material up to about 50,000 years old; for older samples, potassium-argon (T½ = 1.25 billion years) takes over.
Bohr hydrogen — quantized orbits
The simplest quantum atom: an electron allowed in only certain orbits with energy En = −13.6/n² eV (n = 1, 2, 3, ...). Transitions between orbits release or absorb photons with energy equal to the difference. The default sample (ni = 3 → nf = 2, the Hα line) gives ΔE ≈ 1.889 eV and a wavelength of ≈ 656.5 nm, the deep-red Hα line — the same wavelength that colors emission nebulae pink in photographs. Transitions to n = 1 make the Lyman series (ultraviolet), to n = 2 the Balmer series (visible), to n = 3 the Paschen series (near infrared), and so on. The Bohr model is wrong in detail (the real hydrogen atom is described by Schrödinger's equation, with probability clouds instead of orbits), but right enough to predict the hydrogen spectrum exactly and serve as the bridge from classical to quantum thinking.
Common misconceptions
- Time dilation is "just perspective". The dilated time is physical, not apparent. A muon created by cosmic rays high in the atmosphere lives long enough (in our frame) to reach the ground because its high γ extends its lifetime by that factor. The muon itself sees the atmosphere contracted, with the same physics — both views agree on what reaches the ground.
- E = mc² converts mass to energy. Mass does not "become" energy; it is energy. The equation says mass and energy are the same quantity measured in different units. A nucleus lighter than its constituents by Δm has released Δm·c² of binding energy; the energy was always there, just not as mass.
- Light is "either a wave or a particle". Neither alone. Light behaves as a wave in interference and as a particle (photon) in emission/absorption. The two descriptions are complementary, not contradictory.
- Relativistic mass increases with speed. Modern usage has dropped the concept: only rest mass m0 is "mass". Total energy grows with γ (E = γm0c²), but calling that "relativistic mass" obscures the geometry of 4-momentum and is no longer standard.
Related tools: Circuits for the classical electrical background, Thermodynamics for statistical mechanics and the equipartition theorem, and Constants Reference for the values of every constant used here.
