Quantum mechanics is the physical framework used to describe matter and radiation, particularly at atomic and subatomic scales. It explains the structure of atoms, their interactions with light, and phenomena that classical mechanics cannot adequately describe. Rather than assigning every physical quantity a definite value along a predetermined trajectory, it represents systems through quantum states and predicts probabilities for measurement outcomes. Its principles also apply to collective, macroscopic phenomena and underpin important technologies. (ocw.mit.edu)
Historical development
The origins of quantum mechanics lie in difficulties encountered by nineteenth-century physics. In 1900, Max Planck introduced discrete energy elements in his treatment of blackbody radiation. This established energy quantization and the constant (h), now called the Planck constant. In 1905, Albert Einstein proposed light quanta to explain phenomena including the photoelectric effect. Niels Bohr subsequently introduced an atomic model with quantized energy levels in 1913. (nobelprize.org)
Modern quantum mechanics took shape in 1925–1926. Werner Heisenberg, Max Born, and Pascual Jordan developed matrix mechanics, while Erwin Schrödinger formulated wave mechanics. These mathematically different approaches proved equivalent. Born supplied the probabilistic interpretation of the wave function, connecting the mathematical formalism with experimental outcomes. The resulting theory replaced classical electron orbits with quantum states and provided a systematic account of atomic spectra. (nobelprize.org)
Quantum states and measurement
A pure quantum state is represented by a normalized vector in a complex Hilbert space, with vectors differing only by an overall phase describing the same physical state. In a position representation, the state is expressed as a wave function, commonly written (\psi(\mathbf{x},t)). More general states, including statistical mixtures and subsystems entangled with their surroundings, require a density matrix. (preskill.caltech.edu)
The principle of quantum superposition permits linear combinations of state vectors. Crucially, superposition is not simply ignorance about which alternative already exists: relative phases between components affect observable interference. When alternative paths remain coherent, their probability amplitudes combine before probabilities are calculated. This distinction accounts for interference patterns in experiments involving individual particles. (live.ocw.mit.edu)
The Born rule connects states with measurement probabilities. For a normalized position wave function, (|\psi(\mathbf{x},t)|^2) is a probability density, so its integral over a region gives the probability of detecting the particle there. In the standard treatment of ideal measurements, observables correspond to self-adjoint operators. Discrete possible outcomes are their eigenvalues, and projections onto the associated eigenspaces determine outcome probabilities. An ideal projective measurement also specifies a corresponding update of the state. (ocw.mit.edu)
Dynamics, quantization, and uncertainty
For a closed system, state evolution is governed by the Schrödinger equation: [ i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle =\hat H|\psi(t)\rangle, ] where (\hbar=h/(2\pi)), and (\hat H) is the Hamiltonian operator, representing total energy. This evolution is unitary: it preserves normalization and is reversible. Thus, deterministic evolution of a state coexists with probabilistic predictions for measurements. (ocw.mit.edu)
Quantization does not mean that every measurable quantity always occurs in discrete steps. Operators can have discrete spectra, continuous spectra, or both. Confined systems commonly possess discrete energy levels, whereas a free particle can have a continuous energy spectrum. Boundary conditions and the form of the interaction determine the available states. Transitions between atomic energy levels account for characteristic emission and absorption frequencies. (ocw.mit.edu)
The uncertainty principle constrains the statistical spreads of incompatible observables. For position and its corresponding momentum, [ \Delta x,\Delta p\geq\frac{\hbar}{2}. ] Here the uncertainties are standard deviations in a specified state, not merely instrumental errors. The relation follows from the noncommutativity of the position and momentum operators. It should therefore be distinguished from the separate question of how a particular measurement disturbs a system. (ocw.mit.edu)
Quantum tunneling is another consequence of wave dynamics. A particle can have a nonzero probability of crossing a finite potential barrier even when its energy is below the barrier height, unlike the corresponding classical particle. This does not require violating energy conservation; it follows from solving the quantum equations across the barrier. (preskill.caltech.edu)
Entanglement and interpretation
Quantum entanglement occurs when a composite system’s state cannot be described as independent states of its components; for mixed states, the broader criterion excludes mixtures of product states. Entangled systems exhibit correlations that can violate Bell inequalities. Bell’s theorem establishes limits on explanations based on local hidden variables under its assumptions. These correlations do not permit controllable faster-than-light communication. (preskill.caltech.edu)
Quantum decoherence results when interactions with an environment suppress observable interference between certain alternatives in a subsystem. It helps explain why many large-scale systems behave approximately classically. However, explaining the appearance of classical behavior is distinct from resolving the measurement problem: how definite observed outcomes relate to quantum evolution. Interpretations—including Copenhagen-associated approaches, many-worlds accounts, and pilot-wave theories—offer different descriptions of states and measurement. (preskill.caltech.edu)
Applications and scope
Quantum mechanics supplies the theoretical basis for semiconductors, lasers, and superconductivity. It also underlies proposals and implementations of quantum computers, which manipulate quantum information to perform particular computational tasks. Quantum computation does not imply a general speed advantage for every problem; its benefits depend on the task and algorithm. (preskill.caltech.edu)
Nonrelativistic quantum mechanics is appropriate when relativistic effects and particle creation can be neglected. Quantum field theory extends the framework to fields and variable particle numbers, providing the setting for relativistic particle physics. Classical mechanics remains an effective approximation in suitable regimes, rather than being discarded wherever a quantum description is possible. (nobelprize.org)