Mathematical formulation of quantum mechanics

A framework representing quantum states as vectors or density operators, observables as operators, and measurement probabilities through mathematical rules. Its standard nonrelativistic form uses complex Hilbert spaces.

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Hilbert space: A complete inner-product vector space that provides the standard setting for quantum states. Quantum state vectors inhabit Hilbert spaces, where inner products determine transition amplitudes.
Schrödinger equation: A differential equation governing the time evolution of a quantum state under a specified Hamiltonian. It gives the standard continuous-time law for closed-system state evolution.
Matrix mechanics: Heisenberg’s 1925 formulation of quantum theory using arrays of transition amplitudes and noncommuting quantities. It supplied an early algebraic route to the modern operator framework.
Path integral formulation: A formulation of quantum theory that computes amplitudes by summing contributions from possible histories. It reaches the same predictions as operator mechanics through a different mathematical route.
Quantum harmonic oscillator: A quantum system with a quadratic potential and equally spaced energy levels. Its exactly solvable states illustrate operators, spectra, and Hilbert-space bases.
Quantum state: A mathematical representation of a quantum system’s preparation, expressed by a state vector or density operator. States are the central objects the framework represents and evolves.
Quantum measurement: A physical interaction that produces an outcome described probabilistically by quantum theory. Measurement rules connect abstract observables to recorded outcomes.
Wave mechanics: Schrödinger’s formulation of quantum theory using wavefunctions and differential equations. It offered a state-function formulation later connected to Hilbert-space methods.
Phase-space formulation of quantum mechanics: A formulation using functions on classical phase space, including Wigner distributions and star products. It recasts quantum states and dynamics without relying solely on state vectors.
Angular momentum operator: An operator encoding rotational angular momentum, with quantized components and characteristic commutation relations. It demonstrates how symmetries and noncommuting observables appear in the formalism.