Quantum mechanics is the mathematical theory used to describe the behaviour of physical systems at microscopic scales and in situations where quantum effects are important. It describes systems using quantum states and predicts the probabilities of possible measurement outcomes. Unlike classical mechanics, which describes objects using variables such as position and momentum along trajectories, quantum mechanics represents physical possibilities through mathematical states that can exhibit superposition and interference.
The theory applies to electrons, atoms, molecules, photons and many other systems and also forms the conceptual foundation for more advanced quantum theories. The central mathematical objects can include state vectors in Hilbert space, wave functions in position or momentum representations, operators corresponding to observables and density matrices for statistical or mixed states. Quantum mechanics is therefore both a physical theory and a highly developed mathematical framework.
The development of quantum mechanics occurred primarily during the early twentieth century after earlier quantum ideas had exposed limitations in classical physics. Max Planck introduced energy quantisation in 1900, Albert Einstein introduced the photon concept in his explanation of the photoelectric effect, and Niels Bohr proposed quantised atomic states. Louis de Broglie proposed that matter could exhibit wave characteristics. Werner Heisenberg developed matrix mechanics in 1925, while Erwin Schrodinger developed wave mechanics in 1926.
Max Born provided the probabilistic interpretation of the wave function, and Paul Dirac developed a powerful abstract formulation that unified important features of the emerging theory. These approaches were shown to be mathematically equivalent within their applicable framework. The resulting theory replaced the older quantum models based on ad hoc rules with a systematic mathematical structure.
A quantum mechanical system is represented by a state. For a pure state, the state can be represented by a vector in a Hilbert space, conventionally written using Dirac ket notation. A wave function is one representation of such a state, particularly useful for particles described in position space. The wave function itself is generally a complex valued mathematical object and is not ordinarily interpreted as a directly measurable physical density.
The Born rule states that the squared magnitude of an appropriate wave function gives a probability density for measurement outcomes in a position representation. Normalisation ensures that the total probability of all possible outcomes is one. Different mathematical representations can describe the same physical state, and global phase factors do not change observable predictions. These features distinguish quantum states from ordinary classical probability distributions.
Observables in quantum mechanics are represented mathematically by operators, with measurable quantities associated with appropriate self adjoint operators in the standard formalism. A measurement of an observable produces one of its allowed outcomes, which are related to the operators eigenvalues. The state determines the probabilities of obtaining those outcomes through the Born rule. If a state is an eigenstate of an observable, a measurement of that observable produces the corresponding eigenvalue with certainty under the idealised measurement framework.
If the state is a superposition of eigenstates, multiple outcomes can have nonzero probabilities. This structure explains quantised measurements without requiring every physical quantity to have discrete values in every system. For example, a free particle can have a continuous range of possible momentum values, whereas a particle confined by a potential may possess discrete energy eigenvalues.
Time evolution in ordinary nonrelativistic quantum mechanics is governed by the Schrodinger equation. For a closed system with Hamiltonian H, the equation specifies how the quantum state changes with time. The Hamiltonian represents the total energy operator and commonly contains kinetic and potential energy terms. The time evolution generated by the Hamiltonian is deterministic at the level of the quantum state, even though individual measurement outcomes are generally probabilistic. This distinction is important.
Quantum mechanics does not simply say that states randomly jump between arbitrary configurations. Between measurements or other interactions, the state evolves according to a precise mathematical law. Measurement introduces probabilistic outcomes according to the state and the observable being measured, while more sophisticated formulations describe measurement as physical interaction with an apparatus and environment.
Quantum mechanics explains interference because probability amplitudes can combine constructively or destructively. In a two path experiment, for example, a quantum amplitude associated with one possible path can interfere with an amplitude associated with another. The resulting probability is determined by the squared magnitude of the total amplitude, so cross terms can appear that would be absent from an ordinary classical mixture of probabilities.
This is why experiments such as the double slit experiment produce interference patterns even when particles are sent through the apparatus individually. Superposition does not mean that a classical object literally occupies two ordinary macroscopic locations in the everyday sense. It means that the quantum state contains coherent amplitudes associated with alternatives, and subsequent measurements can reveal interference between them when coherence is maintained.
Quantum mechanics also explains entanglement, tunnelling, atomic spectra, molecular structure, spin and the behaviour of many condensed matter systems. When systems interact, their combined quantum state can become nonseparable, producing correlations that cannot be represented as independent states of each subsystem. Quantum tunnelling allows a particle to have a nonzero probability of appearing across a classically forbidden potential barrier. Spin gives particles intrinsic angular momentum with quantised measurement outcomes and has no exact classical counterpart. Identical particles require special quantum statistics, with fermions obeying antisymmetric state requirements and bosons obeying symmetric requirements. These principles lead to the Pauli exclusion principle, Fermi gases, Bose condensation and the structure of matter.
Quantum mechanics has been interpreted in several different philosophical ways, including Copenhagen style interpretations, many worlds approaches, objective collapse theories and hidden variable theories such as Bohmian mechanics. These interpretations differ about questions concerning measurement, wave function collapse, branching, ontology and the meaning of probability, while many reproduce the same standard experimental predictions in ordinary circumstances. It is important to distinguish the mathematical theory from interpretations of what the mathematics means.
Quantum mechanics is extraordinarily successful experimentally, but it also has boundaries. Nonrelativistic quantum mechanics is not the complete theory for situations requiring special relativity and quantum fields. Quantum field theory is needed for particle creation and annihilation and relativistic quantum processes, while a complete theory of quantum gravity remains an open problem.
