Welcome to the Lab: What Is Quantum Mechanics?

A physical theory is a mathematical framework that turns a description of a situation into testable predictions. Quantum mechanics is the framework that governs matter and light at the scale of atoms and below. It specifies which states a system can occupy, how they change in time and, crucially, the probabilities of the results of measurements. That word, probabilities, is the first shock. Classical physics, meaning Isaac Newton's mechanics (1687) and James Clerk Maxwell's electromagnetism (1860s), assumes that quantities vary smoothly and that objects have definite properties whether or not anyone checks. It remains superb for bridges, planets and baseballs.

Between 1900 and 1913 that picture failed in specific, embarrassing ways. It predicted that a hot oven should blaze with unlimited ultraviolet energy; it could not explain why dim blue light ejects electrons from sodium while blinding red light ejects none; and it implied that atoms should collapse within a fraction of a nanosecond (Chapters 3 to 5 examine each crack). Between 1925 and 1927, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, Wolfgang Pauli and others built a replacement, and in the century since, no experiment has contradicted the predictions of quantum theory.

Its home territory is atoms, roughly 10⁻¹⁰ metres across; the electrons that surround them; photons, the indivisible packets of light; and atomic nuclei, tens of thousands of times smaller than their atoms. Yet as far as anyone can tell, the rules apply to everything; what changes with size is whether their signature effects survive long enough to be noticed. The solidity of the floor, the green of a leaf, the shining of the Sun and every chip in your phone are quantum phenomena viewed from far away. The rules of the very small run everything you can see.

To see where the quantum takes over, we need scientific notation: a number between 1 and 10 multiplied by a power of ten, where the exponent counts how many times you multiply or divide by ten. So 10³ is a thousand and 10⁻³ is a thousandth. Strap in. Start with a person about 1.7 metres tall. Shrink about two hundred thousand times and you reach a red blood cell, some 8 micrometres across. Nearly a hundred times smaller is a virus, around 100 nm; a thousand times smaller again, an atom, about 0.1 nm. Inside sits a nucleus only about 2 to 15 femtometres wide, one femtometre being 10⁻¹⁵ metres.

Keep going. Protons and neutrons are built from quarks, which have no measured size at all: if they have any extent, experiments put it below about 10⁻¹⁸ metres. Far beneath that lies the Planck length, about 1.6×10⁻³⁵ metres, assembled from Planck's constant, the speed of light and the gravitational constant. It marks the scale at which physicists expect gravity itself to need a quantum description, although nobody has shown that space is grainy there. From your height down to the Planck length is about 35 powers of ten; from an atom, about 25.

A reliable rule of thumb for where quantum behaviour takes over borrows an idea from Chapter 6: every moving object has a de Broglie wavelength, equal to Planck's constant divided by its momentum (mass times velocity). When that wavelength is comparable to the region the object is confined in, or to the openings it passes through, quantum effects dominate. The electron in a hydrogen atom has a typical wavelength of about 0.33 nm, comparable to the atom itself, so atoms are thoroughly quantum. A 70-kilogram person strolling at 1.4 metres per second has a wavelength near 7×10⁻³⁶ metres, hopelessly beyond detection. Light is always emitted and absorbed in whole photons, though radio photons are too feeble and numerous for the graininess to show.

In October 1900 Max Planck found a formula that matched the light emitted by hot objects, and on 14 December 1900 he presented its derivation to the German Physical Society in Berlin. That derivation forced him to treat the energy of the vibrating charges that emit and absorb light of frequency f (oscillations per second, measured in hertz) as coming only in whole packets of size E = hf, where h is a new constant of nature, now called Planck's constant. Planck regarded the packets as a mathematical device; in 1905 Einstein proposed that light itself really comes in such lumps (Chapters 3 and 4). Read E = hf as an exchange rate: h converts a frequency into the energy of one quantum.

Its value is h = 6.62607015×10⁻³⁴ J·s, exactly. The joule-second is energy multiplied by time, a combination called action, so h is often called the quantum of action. Since 20 May 2019 this number has been fixed by definition in the International System of Units, and the kilogram is now defined through it. The smallness of h is why the quantum world hides: one photon of green light at 532 nm carries only 3.7×10⁻¹⁹ joules, so even a 1-milliwatt laser pointer fires about 2.7×10¹⁵ photons per second. Graininess that fine looks smooth. I had this number engraved on Psigor's chassis.

Its close relative appears on almost every page of this course: the reduced Planck constant, ħ, defined as ħ = h/2π, about 1.055×10⁻³⁴ J·s. The 2π appears because physicists often measure oscillation by angle rather than by count. One cycle is 2π radians, so the angular frequency ω equals 2πf, and Planck's relation becomes E = ħω. The uncertainty principle, electron spin and the Schrödinger equation are all written with ħ. Loosely speaking, imagining h shrinking towards zero turns quantum predictions back into classical ones.

This course is built around five big ideas. The first is quantization: certain quantities, above all the energy of anything bound or confined, take only specific, separated values, like the rungs of a ladder. Measuring the energy of hydrogen's electron can give −13.6 eV or −3.4 eV, but never anything in between; the minus signs mean the electron is bound. Jumping between rungs by light means emitting or absorbing a photon whose energy matches the gap. That is why every element, heated as a thin gas, glows with its own barcode of spectral lines, the fingerprints astronomers use to read the stars. A free electron, by contrast, can have any energy; the ladder appears when a wave is confined.

The second is wave-particle duality. Light and matter travel like waves, spreading out and interfering so that crests reinforce or cancel, yet they are always detected like particles, as single localized clicks or dots. Electrons sent one at a time through two slits land as individual dots, but thousands of dots build a striped interference pattern that forms only if each electron's wave passes through both slits. Wave and particle are classical caricatures of one quantum object that is neither.

The third is superposition: a quantum state can be a weighted combination of other states. An electron's spin can be in a combination of up and down; a photon, of taking the left slit and the right. The weights are complex numbers called probability amplitudes, and each outcome's probability is the squared magnitude of its amplitude, the Born rule (Chapter 9). Superposition is not ignorance, like a coin hidden under a cup: superposed possibilities interfere with one another, and that interference drives everything from the double-slit pattern to quantum computing.

The fourth is uncertainty. No quantum state has both perfectly sharp position and perfectly sharp momentum; Heisenberg's principle says their spreads obey Δx·Δp ≥ ħ/2. This is not about clumsy instruments: a wave squeezed into a small region must be built from many wavelengths, each corresponding to a different momentum. It is why atoms have a size at all. The fifth is entanglement: two or more particles can share one joint state that cannot be split into separate descriptions of each, and their measured correlations can exceed anything pre-arranged local instructions could produce, as Bell's theorem predicted and the experiments honoured by the 2022 Nobel Prize in Physics confirmed. Yet entanglement cannot carry a message faster than light.

People sometimes dismiss quantum mechanics as just a theory. In science, though, a theory is not a hunch but a well-tested framework whose quantitative predictions could have failed and did not. Quantum electrodynamics, or QED, the quantum theory of light interacting with charged matter, is often called the most precisely tested theory in science, thanks to the magnetism of a single electron. Every electron behaves like a tiny bar magnet whose strength is summarised by a number called the g-factor.

In 1928 Paul Dirac's relativistic equation for the electron predicted that g should be exactly 2. In 1947 Polykarp Kusch and Henry Foley at Columbia University found it slightly larger, by about a tenth of a percent. In 1948 Julian Schwinger showed that QED accounts for the excess: the electron constantly interacts with the quantum electromagnetic field, and the leading correction to g/2 equals α/2π, about 0.00116, where α, the fine-structure constant (about 1/137), sets the strength of electromagnetism. Kusch shared the 1955 Nobel Prize for the measurement; Schwinger, Richard Feynman and Sin-Itiro Tomonaga shared the 1965 prize for QED.

In 2023, Gerald Gabrielse's group at Northwestern University, studying a single electron in a trap cooled to a fraction of a degree above absolute zero, reported g/2 = 1.00115965218059 with an uncertainty of 0.13 parts per trillion. Toichiro Kinoshita and colleagues evaluated more than twelve thousand Feynman diagrams to push the prediction to similar depth. Theory and experiment agree to about one part in a trillion, like measuring the distance to the Moon to within a few sheets of paper; the main limit now is how well we know α. Science is the only madness with receipts.

Now the other half of the myth: that nobody uses it. You are using it right now. Computer chips are built from transistors, switches that let a small voltage control a larger current. Quantum mechanics shows that electrons in a crystal occupy bands of energy separated by forbidden gaps, and how those bands fill decides whether a material is a metal, an insulator or a semiconductor such as silicon, a picture built by Felix Bloch (1928) and Alan Wilson (1931). At Bell Labs in December 1947, John Bardeen and Walter Brattain, in William Shockley's group, made the first working transistor; all three shared the 1956 Nobel Prize. Today's chips pack billions, with insulating layers so thin that engineers must design around electrons tunnelling through them.

Light-based technology is just as quantum. In 1917 Albert Einstein predicted stimulated emission: a passing photon can trigger an energized atom to emit a second photon that is its identical twin, with the same direction, frequency and phase. On 16 May 1960, at Hughes Research Laboratories in California, Theodore Maiman turned that stimulated emission into the first working laser, using a ruby crystal. An LED, or light-emitting diode, lets electrons fall across a semiconductor's band gap, each releasing a photon whose energy, and so colour, is set by the gap through E = hf. Efficient blue LEDs, which made white LED lighting possible, earned Isamu Akasaki, Hiroshi Amano and Shuji Nakamura the 2014 Nobel Prize in Physics.

Medicine and navigation lean on quantum rules too. The protons in your body's water carry spin, an intrinsic quantum magnetism (Chapter 17); in a 1.5-tesla magnetic resonance imaging (MRI) scanner they absorb and re-emit radio waves at about 64 MHz, signals mapped into images of soft tissue. GPS satellites carry atomic clocks that tick at the frequency of a transition between two quantized energy levels, the same in every atom of that isotope; the SI second itself is defined as the duration of exactly 9,192,631,770 cycles of the radiation from one such transition in caesium-133. Radio signals travel about 30 centimetres per nanosecond, so precise timing becomes precise position. In a solar cell, each absorbed photon carrying at least silicon's band-gap energy of 1.12 eV lifts one electron across the gap, a photon-by-photon process explained by Einstein's light quantum (Chapter 4).

You need three mathematical tools. Scientific notation you have already met: to multiply, multiply the leading parts and add the exponents, so (2×10³) × (3×10⁻⁵) = 6×10⁻². Probability is a number between 0 and 1 giving how often an outcome occurs over many identical trials. Quantum mechanics predicts probabilities: it can say each photon has a 30 per cent chance of passing a filter, so about 300,000 of a million will pass, but not which ones; as far as any experiment can tell, single outcomes are random. Complex numbers, built on i, the number whose square is −1, are best pictured as arrows on a plane; adding them means adding arrows, and interference is arrows partly cancelling. Chapter 2 introduces them gently.

Every chapter follows the same rhythm. Start with the video briefing at the top. Then read the lesson, or press play and let the narrator read the identical words aloud, highlighting each passage. Next come the key equations in plain words, worked examples, myth-busting and memory aids, from a mnemonic to flashcards. Then play the lab, a game that puts the idea in your hands, and finish with the printable worksheet and the Quiz Show. Revisit the flashcards after a day, three days and a week. And for this week's discoveries, the daily What's New in Quantum bulletin awaits.