Quantum mechanics

• 1. Introduction

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  Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light, and its strangest features show up at and below the scale of atoms. Richard Feynman, one of its great practitioners, once said something that should worry anyone who claims mastery of it. "I think I can safely say that nobody understands quantum mechanics." Steven Weinberg put a similar feeling more bluntly, stating that in his opinion there is now no entirely satisfactory interpretation of the theory.

  This is a theory that works. Its predictions have been verified to an extraordinary degree of accuracy. One refinement, the theory of how light and matter interact, agrees with experiment to within 1 part in a trillion when predicting the magnetic properties of an electron. And yet the people who use it most cannot agree on what it means. How can a description of nature be that precise and that mysterious at once? Why can it usually tell you only the odds of an outcome, never the outcome itself? And how did a problem about glowing hot objects in 1900 grow into the foundation of all quantum physics?

• 2. The Rule Of Probabilities

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  A quantum particle like an electron can be described by a wave function, a mathematical object that assigns to every point in space a probability amplitude. To turn that amplitude into something you can test, you take the square of its absolute value. This is the Born rule, named after the physicist Max Born, and it converts amplitudes into a probability density for where the electron will be found.

  The Schrodinger equation ties one moment to the next. It relates the collection of probability amplitudes at one instant to the collection at another, and its solutions can be eigenstates of the Hamiltonian, which produce probability distributions that do not change with time. A single electron in an unexcited atom is pictured classically as a particle circling the nucleus. In quantum mechanics it is instead a static wave function surrounding the nucleus. For an unexcited hydrogen atom that wave function is spherically symmetric and is called an s orbital.

  Closed-form solutions to the Schrodinger equation are rare. They are known for the quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the hydrogen atom. Even the helium atom, with just two electrons, has defied every attempt at a fully analytic treatment. Where exact answers fail, physicists reach for perturbation theory, which starts from a simple model and adds a weak potential to approximate a harder one.

• 3. Tradeoffs And Interference

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  No preparation of a quantum particle can give precise predictions for both its position and its momentum at the same time. That is the most familiar form of the uncertainty principle, and it follows directly from the basic formalism. Position and momentum are observables represented by Hermitian operators, and these two operators do not commute. Either standard deviation can be made arbitrarily small on its own, but never both together.

  The double-slit experiment puts another quantum effect on display. A coherent light source such as a laser beam illuminates a plate pierced by two parallel slits, and a screen behind catches the light. Bright and dark bands appear, the signature of waves interfering, a pattern you would never expect from classical particles. Yet the light always lands on the screen at discrete points, like individual particles, and the pattern builds up from the density of those hits. Put detectors at the slits and each photon is found to pass through one slit, not both, and the interference vanishes. This is wave-particle duality, and electrons, atoms, and molecules all show it.

  A particle can also cross a barrier it lacks the energy to climb. This is quantum tunnelling, and in classical mechanics the particle would simply be trapped. Tunnelling enables radioactive decay and the nuclear fusion that powers stars, and it underlies devices like the scanning tunnelling microscope, the tunnel diode, and the tunnel field-effect transistor.

• 4. The Trait That Departs

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  Erwin Schrodinger called it "the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought." He was describing entanglement, the state in which two systems become so intertwined that the whole can no longer be described in terms of its separate parts. When two systems combine, their joint space is the tensor product of the two component spaces, and some valid joint states simply cannot be written as a product of separate states. Those are the entangled ones.

  Describing one half of an entangled pair on its own requires a reduced density matrix, which captures the statistics of measurements on that piece alone. That comes at a cost. Knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the whole. Entanglement also drives quantum decoherence, in which a system becomes entangled with its environment, which helps explain why quantum effects are so hard to see in anything larger than the microscopic.

  Despite a popular misconception, entanglement does not let you send signals faster than light, a limit fixed by the no-communication theorem. What it does power is quantum computing and quantum communication protocols such as quantum key distribution and superdense coding. It also opened a way to test for hidden variables, properties more fundamental than quantum theory itself that might allow sharper predictions. Bell's theorem showed that broad classes of local hidden-variable theories are incompatible with quantum physics, and many Bell tests have returned results that violate the constraints such theories would impose.

• 5. The Argument Over Meaning

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  Perhaps the only consensus about the foundations of quantum mechanics is that there is no consensus. The debates centre on the theory's probabilistic nature, on the trouble with wave function collapse and the related measurement problem, and on quantum nonlocality. The probabilistic nature stems from the act of measurement itself, which makes measurement one of the most contested ideas in all of physics.

  The views of Niels Bohr, Werner Heisenberg, and others are grouped as the Copenhagen interpretation. On this reading, the randomness of the theory is not a temporary gap waiting for a deterministic replacement, but a final break with the classical idea of causality. Bohr stressed that any use of the formalism must refer to the experimental arrangement, because evidence gathered under different setups is complementary. Copenhagen-type views were held by Nobel laureates including Bohr, Heisenberg, Schrodinger, Feynman, and Zeilinger.

  Albert Einstein, himself a founder of quantum theory, could not accept its apparent disregard for determinism and locality. In 1935 he, Boris Podolsky, and Nathan Rosen argued that the principle of locality implies quantum mechanics is incomplete, a thought experiment now called the Einstein-Podolsky-Rosen paradox. In 1964 John Bell showed that locality together with determinism was actually incompatible with quantum mechanics, and later experiments violated his Bell inequalities, falsifying that conjunction. Other interpretations went further still. Bohmian mechanics restores determinism by making the theory explicitly nonlocal and adding a real position guided by a nonlocal equation, with no collapse at all.

• 6. Many Worlds

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  Everett's many-worlds interpretation, formulated in 1956, holds that every possibility described by quantum theory occurs at once, across a multiverse of mostly independent parallel universes. It follows from a single move: removing the axiom that the wave packet collapses. Every possible state of the measured system, the measuring apparatus, and the observer is present together in one real physical superposition.

  The multiverse, on this account, is deterministic. We perceive randomness governed by probabilities only because we never observe the whole multiverse, just one parallel universe at a time. Exactly how that perception arises has been argued at length, and several attempts to derive the Born rule from the picture have reached no agreement on whether they succeed.

  Newer ideas grew from the same soil. Relational quantum mechanics appeared in the late 1990s as a modern descendant of Copenhagen-type thinking, and QBism was developed some years after that.

• 7. From Black Bodies To Solvay

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  The black-body radiation problem was discovered by Gustav Kirchhoff in 1859, and it would not yield to classical physics. In 1900 Max Planck proposed that energy is radiated and absorbed only in discrete packets he called quanta, a calculation that matched the observed radiation precisely. The word quantum comes from the Latin for "how great" or "how much." Planck was cautious, treating his hypothesis as a mathematical trick rather than a real feature of radiation.

  In 1905 Albert Einstein took Planck's quanta literally and used them to explain the photoelectric effect, in which light shone on certain materials ejects electrons. Niels Bohr then built Planck's ideas into a model of the hydrogen atom that predicted its spectral lines, and Einstein went on to show that light itself could be described as a particle, later called the photon. This loose collection of fixes became known as the old quantum theory, never complete or self-consistent, a set of heuristic corrections to classical mechanics now understood as a semi-classical approximation.

  The modern theory arrived in a rush. In 1923 Louis de Broglie proposed his theory of matter waves, that particles can show wave characteristics and the reverse. In 1925 Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics while Erwin Schrodinger invented wave mechanics, and in July 1926 Born introduced the probabilistic reading of the wave function. The field won wider acceptance at the Fifth Solvay Conference in 1927. By 1930 David Hilbert, Paul Dirac, and John von Neumann had unified and formalized it, sharpening the focus on measurement and on the statistical nature of what we can know about reality.

• 8. Where The Theory Reaches

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  Solid-state physics and materials science depend on quantum mechanics, which is often the only theory that can reveal how individual subatomic particles behave: electrons, protons, neutrons, photons, and others. Much of modern technology operates at scales where quantum effects matter. The list of applications runs through quantum chemistry and quantum optics, superconducting magnets, light-emitting diodes, the optical amplifier and the laser, the transistor and the semiconductors behind the microprocessor, and imaging methods like magnetic resonance imaging and electron microscopy. The macro-molecule DNA itself is rooted in the nature of the chemical bond.

  Many everyday properties of bulk matter trace back to quantum behavior. The stability of matter, the rigidity of solids, and the thermal, optical, and magnetic properties of materials all follow from electric charges interacting under quantum rules. At temperatures near absolute zero, quantum behavior can even show itself on a macroscopic scale, in superconductors and superfluids.

  One frontier still resists. The formalisms of quantum theory and general relativity contradict each other, and uniting them has been a major goal of 20th and 21st-century physics. String theory replaces point-like particles with one-dimensional strings, one of whose vibrational states corresponds to the graviton. Loop quantum gravity instead weaves space from finite loops called spin networks, whose evolution over time is a spin foam. Its characteristic scale is the Planck length, roughly 1.616 times ten to the minus thirty-five metres, below which lengths carry no physical meaning in the theory.