Quantum mechanics

Quantum mechanics is the fundamental scientific theory that explains the behaviour of matter and radiation at atomic and subatomic scales. It is the theoretical foundation underlying all branches of quantum physics, including quantum chemistry, quantum field theory, quantum information science and modern quantum technologies. Many microscopic phenomena cannot be accounted for by classical physics, which remains an approximation valid primarily at macroscopic and ordinary microscopic scales. Quantum mechanics replaces the classical description with a probabilistic framework in which observable quantities are quantised and the outcomes of measurements are inherently uncertain.

Origins and Historical Development

The emergence of quantum mechanics began with attempts to explain observations that defied classical theory. In 1900 Max Planck introduced the idea of energy quantisation to solve the blackbody radiation problem. In 1905 Albert Einstein explained the photoelectric effect by proposing that light consists of energy quanta. These early ideas, collectively referred to as the old quantum theory, established the groundwork for a more complete framework.
During the mid-1920s a series of major advances by Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Max Born, Paul Dirac and others resulted in the modern formulation of quantum mechanics. Schrödinger introduced wave mechanics, Heisenberg matrix mechanics and Born the probabilistic interpretation. These developments unified earlier insights into a consistent theory. Subsequent progress led to quantum electrodynamics, relativistic quantum theories and the general mathematical formulation in terms of Hilbert spaces.

Fundamental Concepts and Phenomena

Quantum mechanics provides quantitative predictions for the properties and dynamics of particles, atoms, molecules and other microscopic systems. It applies universally, and experiments have confirmed its accuracy to remarkable precision. Even complex molecules containing thousands of atoms have been shown to behave as quantum systems.
A defining feature of quantum mechanics is that it does not generally predict exact outcomes. Instead, it yields probability amplitudes, which are complex numbers whose squared magnitudes give probabilities. This principle, known as the Born rule, forms a core part of quantum theory. For example, an electron in a hydrogen atom is described by a wave function whose squared magnitude provides the probability density for finding the electron in different regions of space. Brighter regions in graphical depictions of atomic orbitals correspond to higher probabilities.
A distinctive quantum property is wave–particle duality. In the double-slit experiment, light or matter forms an interference pattern characteristic of waves, yet detection occurs at discrete points as particles. If detectors identify which slit a particle passes through, the interference pattern disappears, demonstrating that quantum behaviour depends on the measurement context.
Another central idea is quantisation. Quantum systems exhibit discrete values of energy, angular momentum and other quantities. The energy levels of electrons in atoms, for instance, form a set of fixed values rather than a continuum.
Quantum mechanics also introduces the uncertainty principle, which states that certain pairs of observables, such as position and momentum, cannot both be known with arbitrary precision. This limitation arises not from experimental imperfections but from the mathematical structure of the theory.
A further phenomenon is quantum tunnelling, in which a particle can penetrate a potential barrier even though classical mechanics forbids it. Tunnelling is crucial for nuclear fusion in stars, radioactive decay and technologies such as tunnel diodes and scanning tunnelling microscopes.
When quantum systems interact, they may become entangled, meaning that their physical properties can no longer be described independently. Schrödinger identified entanglement as a defining feature of quantum mechanics. Entanglement enables quantum key distribution, quantum computing and superdense coding. Experiments testing Bell’s theorem have shown that local hidden-variable theories cannot reproduce the predictions of quantum mechanics.

The Schrödinger Equation

Central to non-relativistic quantum mechanics is the Schrödinger equation, which determines how the wave function evolves. It relates probability amplitudes at one time to those at another and allows calculation of energy levels, transition probabilities and dynamical behaviour. Solutions to the Schrödinger equation explain the structure of atoms and molecules, the spectra they produce and their interactions with fields.

Mathematical Formulation

The modern formulation of quantum mechanics employs the language of functional analysis. The state of a physical system is represented by a vector ∣ψ⟩|\psi\rangle∣ψ⟩ belonging to a complex separable Hilbert space H\mathcal{H}H, normalised such that ⟨ψ∣ψ⟩=1\langle \psi | \psi \rangle = 1⟨ψ∣ψ⟩=1. States that differ only by multiplication by a complex number of unit modulus represent the same physical state, since such a factor contributes only a global phase.
Observable quantities correspond to linear, self-adjoint operators on H\mathcal{H}H. The possible measurement results are given by the operator’s eigenvalues, and the probability of obtaining a result associated with a given eigenvector is determined by the projection of the state onto that eigenvector. The time evolution of the state is generated by the system’s Hamiltonian operator through the Schrödinger equation.
This mathematical framework underpins a wide variety of applications, from atomic physics and condensed-matter theory to quantum information processing.

Applications and Experimental Tests

Quantum mechanics successfully accounts for the spectra of hydrogen and other atoms, chemical bonding, superconductivity, semiconductors and countless other phenomena. Quantum electrodynamics, an extension describing the interaction of matter with light, has been verified to extraordinary precision, achieving agreement at the level of one part in 101210^{12}1012 for the magnetic moment of the electron.
The theory is also essential for modern technologies such as lasers, transistors and magnetic resonance imaging. Experimental demonstrations of interference, entanglement, tunnelling and quantisation have repeatedly confirmed that quantum mechanics correctly describes nature.

Interpretation and Philosophical Questions

Although its mathematical structure is well established, quantum mechanics raises deep conceptual issues. The question of how measurement produces definite outcomes remains the subject of various interpretations, including the Copenhagen interpretation, many-worlds theory, pilot-wave theory and others. Thought experiments such as Wigner’s friend explore the implications of measurement and observer dependence.
Quantum theory applied to macroscopic or cosmological systems poses further challenges. While it describes molecules of increasing complexity, its extension to observers or the entire universe introduces philosophical questions. Nevertheless, the mathematical formalism remains the foundation for understanding microscopic physics.