Science › Quantum world
The wave function
1926· Theory
ProvenTested many times; part of working physics.
The mathematical object that holds everything that can be known about a quantum system, and how that changes over time.
See it on the Science map →How it works
A wave function assigns a number to each possible outcome, such as each place an electron could be found. Schrödinger's equation says how it evolves, smoothly and predictably. Max Born's 1926 rule turns it into odds: the squared size of the number gives the probability of finding the particle there. It predicts statistics precisely but never which single outcome will occur.
Why it matters
Nearly every quantum prediction, from chemical bonds to transistors, is computed from wave functions. It made physics probabilistic at its foundation.
Think of it like
Like a pro forma for a property: it lays out possible outcomes with weights, but does not tell you which tenant will actually sign.
How we know
Wave-function calculations predict atomic spectra, chemical reactions and electronic behavior with great accuracy. Born's probability rule has passed every statistical test performed.
Still a mystery
Is the wave function physically real, or just a summary of our knowledge? Physicists still split on this.
Connections (8)
▸Erwin Schrödinger wrote the equation for The wave functionProven
The relationship
Schrödinger wrote an equation that tells how a quantum system's wave function changes over time. Feed in the forces, such as the pull of a nucleus, and it yields the allowed energy levels and how waves spread and interfere. It does for quantum systems what Newton's laws do for planets.
How the link was found
Erwin Schrödinger, in four papers in early 1926, begun over a Christmas 1925 holiday in Arosa in the Swiss Alps, inspired by de Broglie's matter waves.
Why it's proven
It reproduced hydrogen's levels without Bohr's ad hoc rules and predicts chemistry and solid-state physics accurately. Max Born's probability interpretation, the same year, made its meaning testable.
Open questions and mysteries
The equation is smooth and predictable, yet measurements give random single outcomes. How the two fit together is the measurement problem.
▸Hilbert space: the stage for quantum states is the math home of The wave functionProven
The relationship
A wave function is a quantum state, and each possible state is a vector in Hilbert space. That framework says how states combine (superposition), how they change over time, and how measurement odds are read off by projecting the vector. Hilbert space unified the rival wave and matrix versions of quantum mechanics into one theory.
How the link was found
Paul Dirac and John von Neumann, about 1927–1932; von Neumann's 1932 book 'Mathematical Foundations of Quantum Mechanics' made it standard.
Why it's proven
Every quantum calculation that matches experiment, from atomic spectra to quantum computers, is done in this framework.
Open questions and mysteries
Whether the vector is physically real or only encodes our knowledge, and whether Hilbert space is the final framework, are open questions.
▸Complex numbers are required by The wave functionProven
The relationship
A wave function gives each possibility a complex number, which carries both a size and a phase (an angle). When possibilities combine, phases can reinforce or cancel, producing interference. Real numbers alone can't track phase this way. The Schrödinger equation itself contains i, so the wave function is unavoidably complex.
How the link was found
Erwin Schrödinger's 1926 equation built i in; he initially hoped to remove it but could not.
Why it's proven
Interference experiments confirm phase effects. Experiments in 2022 ruled out real-number quantum theory, following a 2021 proposal by Marc-Olivier Renou and colleagues.
Open questions and mysteries
Why nature's basic description needs complex numbers rather than real ones is not understood.
▸Probability via the Born rule, extracts predictions from The wave functionProven
The relationship
The wave function itself is never observed directly. The Born rule says the chance of getting a particular result equals the square of the size of the wave function's number for that result. Probability is the bridge from the abstract wave function to what labs actually measure: how often each outcome occurs.
How the link was found
Max Born proposed it in 1926, adding the crucial squaring in a footnote; he received the Nobel Prize for it in 1954.
Why it's proven
Measured outcome frequencies match it in every quantum experiment, including dedicated tests for small deviations, which found none.
Open questions and mysteries
Why the rule takes this form, and whether it can be derived rather than assumed (for example in the many-worlds interpretation), is debated.
▸The wave function describes Superposition: many states at onceProven
The relationship
Schrödinger's equation is linear: if two wave functions are valid solutions, so is any blend of them. So a wave function naturally describes a system as a weighted mix of possibilities, such as passing through both slits at once. The weights set the odds of each outcome when measured.
How the link was found
Built into quantum mechanics from 1926 (Schrödinger, Born); Dirac's 1930 textbook made the superposition principle a founding rule.
Why it's proven
Interference effects, which require weighted blends adding and cancelling, are observed in every quantum system tested.
Open questions and mysteries
Whether the wave function is a real thing in a superposition, or a summary of information, is unresolved.
▸The wave function predicts Quantum tunnelingProven
The relationship
In the wave equation, a particle's wave does not stop abruptly at an energy barrier; it shrinks steadily inside it. If the barrier is thin enough, some wave remains on the far side. The amount remaining gives the probability the particle turns up beyond the barrier.
How the link was found
Friedrich Hund noticed it in molecules in 1927; George Gamow, and separately Ronald Gurney and Edward Condon, used it to explain alpha decay in 1928.
Why it's proven
Predicted alpha-decay half-lives, ranging from tiny fractions of a second to billions of years, match the formula. Tunnel diodes and scanning tunneling microscopes work as predicted.
Open questions and mysteries
How long a particle takes to cross the barrier is still debated, with experiments and theories giving conflicting answers.
▸The wave function forbids losing information, creating The black-hole information paradoxOpen question
The relationship
In quantum mechanics, a system's wave function evolves by a rule that is always reversible, called unitarity. In principle, the present state always determines the past, so information is never truly lost. Hawking's calculation said black holes turn any input into the same featureless heat. That would erase information, violating the rule. The clash between these two claims is the paradox.
How the link was found
Hawking stated the conflict in 1976 in "Breakdown of predictability in gravitational collapse," arguing quantum mechanics must be modified.
Is it proven?
Reversibility is tested in every quantum experiment to date. Whether black holes respect it cannot be tested directly; most physicists now think they do, based on theoretical results such as AdS/CFT.
Open questions and mysteries
If information is preserved, where exactly is it stored and how does it get into the radiation?
▸The wave function is applied to the whole universe in The no-boundary proposal (Hartle–Hawking)Speculative
The relationship
Normally a wave function gives odds for a particle's position. The 'wave function of the universe' gives odds for every possible shape and size of space and its fields. The Wheeler–DeWitt equation says how it must behave, but allows many solutions. The no-boundary proposal picks one: add up only histories that round off smoothly with no edge.
How the link was found
Bryce DeWitt wrote the universe's wave equation in 1967. Hartle and Hawking chose its solution in 1983 using Feynman's sum over histories; Vilenkin proposed a rival choice.
Is it proven?
Untested. The calculations work only in simplified models with a few variables. Whether a wave function for everything makes sense is unclear, since no outside observer exists to measure it.
Open questions and mysteries
With no outside observer, what makes one outcome real? This drove Hartle and Murray Gell-Mann's 'decoherent histories' work from 1990, and feeds interest in many-worlds.