Background
Quantum mechanics is the mathematical foundation of quantum physics, the branch of science that describes the behavior of systems at atomic and subatomic scales.
While Newton’s laws form the basis of classical mechanics, which explains how bridges, planes, and other everyday objects work, quantum mechanics abandons the precise results of those laws for a system based on probability.
This feature comes from simultaneously describing objects as extended waves and localized particles, introducing uncertainty into experimental outcomes (watch explainer).
Understanding quantum phenomena has resulted in numerous technologies, including atomic clocks that maintain GPS systems and the semiconductors in modern electronics.
Quantization
By the turn of the 20th century, scientists understood light could have different frequencies corresponding to different colors within the visible spectrum (e.g., purple is high frequency and red is low frequency).
Just as stovetop rings glow mostly red or orange—not a full rainbow with UV and X-rays—heated objects were observed to radiate most intensely at one frequency and emit little to no radiation at higher and lower frequencies (see visualization).
However, classical mechanics predicted that every heated object should emit high-frequency light with infinite intensity, even though this would require unlimited energy (watch explainer).
To fit observational data, Max Planck suggested that energy must be quantized rather than continuous—that is, it could only exist in discrete, minimum amounts called quanta.
Much like a height limit on a ride does not mean someone half the size can still ride half the roller coaster, quantization sets minimum thresholds for all frequencies of emitted light.
Since nothing reaches the infinite temperature needed to surpass high-frequency thresholds, no such light is emitted, matching observations and averting the catastrophe.
Wave-Particle Duality
Although Planck viewed quantization and thresholds as mathematical conveniences done “in an act of despair,” they proved revolutionary in illuminating various phenomena.
To explain the photoelectric effect, where metals emit electrons only when shined upon by light waves of high enough frequency, Albert Einstein proposed chunks of light with discrete energy. These chunks—particles of light later called photons—could collide with electrons like billiard balls and eject them from atoms (watch explainer).
To explain why different gases emit unique colors when heated, Niels Bohr proposed that electrons orbit the nucleus in discrete energy levels unique to each element. As they jump from higher to lower levels, electrons emit light of specific frequencies (learn more).
Realizing that both Bohr’s electrons and Einstein’s light particles had discrete energies, Louis de Broglie hypothesized that matter might concurrently exist as waves just as light does.
His wave-particle duality explained why discrete atomic energy levels exist and was experimentally verified when matter was shown to interfere and superimpose (as waves do) in tests like the double-slit experiment (explore simulation).
Matrix and Wave Mechanics
Building on de Broglie, Werner Heisenberg developed quantum mechanics—the mathematical language for describing quantum phenomena—using matrices.
Heisenberg used it to show that the wave-particle duality limits how precisely we can simultaneously know a pair of properties in a quantum system, such as a particle’s position and momentum (watch explainer). His uncertainty principle modified Bohr’s model of discrete electron orbits around the nucleus into probability clouds called orbitals (see visualization).
A year after Heisenberg’s work, Erwin Schrödinger developed another version of quantum mechanics using waves. Schrödinger’s wave function describes how a quantum system evolves with time and can be used to determine the odds of experimental outcomes rather than which will definitively occur.
Both matrix and wave mechanics were validated through their accurate modeling of the hydrogen atom. When combined with special relativity, wave mechanics also predicted the existence of antimatter four years before its discovery.
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