Field · Emerged 1924 – 1932

Quantum Mechanics

What are the laws of motion for atoms and electrons, and what do they say about reality?

5 chapters4 min read6 turning points 1 open problem

Branched from: Old Quantum Theory
Branched into: Nuclear Structure + Quantum Field Theory + Quantum Information + Quantum Optics + Solid-State Physics + Stellar Astrophysics
Figures: Louis de Broglie, Werner Heisenberg, Max Born, Pascual Jordan, Erwin Schrödinger, Clinton Davisson, Lester Germer, George Paget Thomson

In brief

Quantum mechanics is the theory of how matter and light behave at the scale of atoms. A particle is described by a wave function that spreads through space, evolves smoothly according to Schrödinger's equation, and gives only the probabilities of what a measurement will find. Quantities like energy come in discrete levels, particles can be in superpositions of several states at once, and some pairs of quantities, such as position and momentum, cannot both be sharp.

It was created in a rush between 1925 and 1927, mostly by physicists in their twenties, and replaced the patchwork of the old quantum theory with a single consistent framework. It explains the periodic table, chemical bonds, the solidity of matter, the behaviour of semiconductors and the light of the stars. It has never failed an experimental test. What it says about reality is still argued over.

Key ideas

Matter wavesEnters 1924: Every particle has a wavelength $\lambda = h/p$ , where $p$ is its momentum. For electrons it is about the size of an atom, which is why atoms behave quantum mechanically.
Wave function and Schrödinger's equationEnters 1926: A particle's state is a wave function, $\psi$ , whose evolution is governed by Schrödinger's equation. Its allowed standing-wave patterns give the discrete energy levels of atoms.
Born ruleEnters 1926: The probability of finding a particle at a place is $|\psi|^2$ there. Quantum mechanics predicts probabilities, not individual outcomes.
Uncertainty principleEnters 1927: Position and momentum cannot both be precisely defined: $\Delta x \, \Delta p \ge \hbar/2$ . It is a property of waves, not a limitation of instruments.
SuperpositionEnters 1926: A quantum system can be in a combination of states, such as passing through two slits at once. Measurement yields one outcome, with probabilities set by the combination.

Draws on other domains

↙ Mathematics
Complex Analysis
Quantum mechanics is written in complex numbers
↙ Mathematics
Numerical Linear Algebra
Energy levels as eigenvalues
↙ Mathematics
Representation Theory
Atomic spectra and spin

Chapter I

Particles as Waves

The old quantum theory had shown that light waves behave like particles. In 1924 Louis de Broglie, a French aristocrat who had turned from history to physics, proposed the reverse: every particle has a wavelength, $\lambda = h/p$ . Bohr's allowed orbits became the ones where a whole number of electron waves fits around the circle. De Broglie's examiners did not know what to make of his thesis and sent it to Einstein, who said it lifted a corner of the great veil.

Chapter II

Two Mechanics

The new theory arrived twice in a year. In June 1925 Werner Heisenberg, aged twenty-three, went to the treeless island of Heligoland to recover from hay fever. There he built a mechanics in which only observable quantities appear, arranged in arrays with a strange rule of multiplication. Max Born recognised the arrays as matrices, which do not commute, and with Pascual Jordan completed the theory.

Over the Christmas holiday of 1925, Erwin Schrödinger found a wave equation for de Broglie's waves. Its solutions for hydrogen gave Bohr's energy levels with no ad hoc rules, as the natural vibration patterns of a wave confined around a nucleus. Schrödinger soon showed that his mechanics and Heisenberg's were mathematically equivalent. Most physicists preferred waves, which they could picture.

Chapter III

Probability and Uncertainty

What was the wave? Schrödinger hoped it was a real spread-out electron. In 1926 Born proposed, in a footnote, that its squared size gives the probability of finding the particle. Physics would predict only the odds of each outcome. In 1927 Heisenberg showed that position and momentum cannot both be sharp. Einstein never accepted that this was the final word, and his debates with Niels Bohr at the Solvay conferences became famous. The same year, Clinton Davisson and Lester Germer at Bell Labs, and George Paget Thomson in Aberdeen, saw electrons diffract exactly as waves should.

Chapter IV

A Closer Look: Why You Don't Diffract

De Broglie's formula $\lambda = h/p$ tells when wave behaviour matters: when the wavelength is comparable to the size of whatever the particle meets. Planck's constant is $h = 6.63 \times 10^{-34}$ J·s.

An electron. In Davisson and Germer's experiment, electrons were accelerated through 54 volts, giving them kinetic energy $E = 54$ eV. Their momentum is $p = \sqrt{2mE}$ , so

\lambda = \frac{h}{\sqrt{2 m E}} = \frac{6.63 \times 10^{-34}}{\sqrt{2 \times 9.11 \times 10^{-31} \times 54 \times 1.60 \times 10^{-19}}} \approx 1.67 \times 10^{-10} \text{ m} ,

0.167 nanometres. The spacing between rows of atoms in nickel is about 0.2 nanometres, so the crystal acts as a diffraction grating, and the electrons emerge in sharp beams at angles set by their wavelength. That is what Davisson and Germer saw.

A baseball. A 145-gram baseball thrown at 40 m/s has momentum $0.145 \times 40 = 5.8$ kg·m/s, so

\lambda = \frac{6.63 \times 10^{-34}}{5.8} \approx 1.1 \times 10^{-34} \text{ m} .

That is less than a ten-billion-billionth of the width of a proton. No slit or grating could ever reveal it. Quantum mechanics applies to baseballs too, but its effects are hidden far below anything measurable.

The same formula explains electron microscopes. Electrons accelerated through 100,000 volts have wavelengths of a few thousandths of a nanometre, far shorter than visible light at 400–700 nanometres, which is why electron microscopes can resolve individual atoms and the structures of proteins.

Chapter V

Success and Unease

Quantum mechanics explained the periodic table, chemical bonding, the conduction of metals and semiconductors, radioactivity and the nuclear fusion that powers the Sun. John von Neumann gave it a rigorous mathematical form in 1932. It has never failed an experimental test. Yet its founders disagreed about what it describes, and the measurement problem, how one definite outcome emerges from a superposition, is still open. Combining it with special relativity led to quantum field theory. Einstein's deepest objection, to entanglement, became the starting point of quantum information.

Turning points6

1924DISCOVERY
De Broglie's matter waves
In his doctoral thesis, Louis de Broglie proposes that if light waves behave as particles, then particles such as electrons should behave as waves, with wavelength $h/p$ . His examiners were unsure what to make of it and sent it to Einstein, who endorsed it. Bohr's allowed orbits became those that fit a whole number of wavelengths.
Louis de Broglie
› Sources (1)
- de Broglie, L. (1925). Recherches sur la théorie des quanta. Annales de Physique (10) 3: 22–128.
1925PARADIGM-SHIFT
Heisenberg's matrix mechanics
Recovering from hay fever on the island of Heligoland, Werner Heisenberg, aged twenty-three, builds a mechanics using only observable quantities, the frequencies and intensities of spectral lines. Max Born recognises its strange multiplication, where $pq \ne qp$ , as matrix multiplication, and with Pascual Jordan makes it a complete theory. It was the first consistent quantum mechanics.
Werner Heisenberg, Max Born, Pascual Jordan
› Sources (2)
- Heisenberg, W. (1925). Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Zeitschrift für Physik 33: 879–893.
- Born, M., Heisenberg, W. & Jordan, P. (1926). Zur Quantenmechanik II. Zeitschrift für Physik 35: 557–615.
1926DISCOVERY
Schrödinger's wave equation
Over the Christmas holiday of 1925 in Arosa, Erwin Schrödinger finds a wave equation for de Broglie's matter waves. Its standing-wave solutions for hydrogen give exactly Bohr's energy levels, now without arbitrary rules. He soon proves that his wave mechanics and Heisenberg's matrices give the same predictions. Physicists found his version far easier to use.
Erwin Schrödinger
› Sources (1)
- Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Erste Mitteilung). Annalen der Physik 79: 361–376.
1926PARADIGM-SHIFTContested
Probability at the foundation
Analysing collisions with Schrödinger's equation, Max Born proposes, in a footnote, that the wave function gives the probability of finding the particle. Physics became fundamentally statistical: the theory predicts only the odds of each outcome. Einstein objected, in a letter to Born, that God does not play dice.
What is disputed
The rule itself is universally used and confirmed. What it means is not agreed. In the Copenhagen view, probabilities are basic and measurement is a primitive notion. In pilot-wave theory, particles have definite positions and the probabilities reflect ignorance. In the many-worlds view, every outcome occurs, and the rule must be derived rather than assumed. These interpretations make nearly identical predictions, and physicists remain divided among them.
Max Born
› Sources (1)
- Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik 37: 863–867.
1927DISCOVERY
The uncertainty principle
Werner Heisenberg shows that the more precisely a particle's position is defined, the less precisely its momentum can be, and argues through thought experiments that no measurement can get around this. Classical determinism, knowing a particle's exact position and velocity, is ruled out in principle.
Werner Heisenberg
› Sources (1)
- Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik 43: 172–198.
1927EXPERIMENT
Electrons diffract like waves
At Bell Labs, Clinton Davisson and Lester Germer scatter electrons from a nickel crystal that had been accidentally recrystallised after an air leak, and find sharp diffraction peaks at the angles de Broglie's wavelength predicts. George Paget Thomson finds the same with thin films in Aberdeen. His father J. J. Thomson had shown the electron to be a particle thirty years earlier.
Clinton Davisson, Lester Germer, George Paget Thomson
› Sources (2)
- Davisson, C. & Germer, L. H. (1927). Diffraction of electrons by a crystal of nickel. Physical Review 30(6): 705–740.
- Thomson, G. P. & Reid, A. (1927). Diffraction of cathode rays by a thin film. Nature 119: 890.

Applications

Where it is used

Structural biology↗ Biology · Molecular Biology
Chemical bonds and the shape of proteins
Quantum mechanics explains why atoms bond and at what angles. Linus Pauling used the quantum theory of the chemical bond to predict the alpha helix of proteins in 1951, before it was seen, and chemists and biologists now compute molecular shapes and reactions from Schrödinger's equation.
› Sources (1)
- Pauling, L., Corey, R. B. & Branson, H. R. (1951). The structure of proteins: two hydrogen-bonded helical configurations of the polypeptide chain. Proceedings of the National Academy of Sciences 37(4): 205–211.
Electronics
Semiconductors and the transistor
Felix Bloch showed in 1928 how electron waves move through a crystal lattice, and band theory followed. It explains why some materials conduct and others insulate, and it made possible the transistor, the integrated circuit and every computer chip.
› Sources (1)
- Bloch, F. (1929). Über die Quantenmechanik der Elektronen in Kristallgittern. Zeitschrift für Physik 52: 555–600.
Technology
Lasers
A laser works by stimulated emission, in which one photon triggers an excited atom to emit an identical one. Theodore Maiman built the first in 1960. Lasers now read barcodes, carry internet traffic in optical fibres and cut steel.
› Sources (1)
- Maiman, T. H. (1960). Stimulated optical radiation in ruby. Nature 187: 493–494.

Open problems

Where the map runs out

Open

The measurement problem

Open as of 2026. Several interpretations are consistent with all experiments.

Schrödinger's equation is deterministic and lets superpositions evolve smoothly. Yet every measurement gives one definite outcome. What counts as a measurement, and what happens to the other possibilities? Schrödinger's cat, both alive and dead until observed, was his way of pointing out how strange the standard answer is.

Why it is hard: Decoherence explains why superpositions of large objects become unobservable in practice, but not why one outcome occurs. The rival interpretations (Copenhagen, many worlds, pilot waves, objective collapse) agree on almost every experiment, so data have little to choose between them. Only collapse theories make different predictions, and experiments are testing them.
What resolving it unlocks: A consistent account of what quantum mechanics describes, which may matter for quantum gravity and for applying the theory to the universe as a whole, where there is no outside observer.

› Sources (2)

Bell, J. S. (1990). Against 'measurement'. Physics World 3(8): 33–40.
Maudlin, T. (1995). Three measurement problems. Topoi 14(1): 7–15.