The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after Erwin Schrödinger, an Austrian physicist, who postulated the equation in 1925 and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

Conceptually, the Schrödinger equation is the quantum counterpart of Newton's second law in classical mechanics. Given a set of known initial conditions, Newton's second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of the wave function, the quantum-mechanical characterization of an isolated physical system. The equation was postulated by Schrödinger based on a postulate of Louis de Broglie that all matter has an associated matter wave. The equation predicted bound states of the atom in agreement with experimental observations.

The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. Other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. When these approaches are compared, the use of the Schrödinger equation is sometimes called "wave mechanics".

The equation given by Schrödinger is nonrelativistic because it contains a first derivative in time and a second derivative in space, and therefore space and time are not on equal footing. Paul Dirac incorporated special relativity and quantum mechanics into a single formulation that simplifies to the Schrödinger equation in the non-relativistic limit. This is the Dirac equation, which contains a single derivative in both space and time. Another partial differential equation, the Klein–Gordon equation, led to a problem with probability density even though it was a relativistic wave equation. The probability density could be negative, which is physically unviable. This was fixed by Dirac by taking the so-called square root of the Klein–Gordon operator and in turn introducing Dirac matrices. In a modern context, the Klein–Gordon equation describes spin-less particles, while the Dirac equation describes spin-1/2 particles.

Introductory courses on physics or chemistry typically introduce the Schrödinger equation in a way that can be appreciated knowing only the concepts and notations of basic calculus, particularly derivatives with respect to space and time. A special case of the Schrödinger equation that admits a statement in those terms is the position-space Schrödinger equation for a single nonrelativistic particle in one dimension: $${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (x,t)=\left[-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)\right]\Psi (x,t).}$$ Here, ${\displaystyle \Psi (x,t)}$ is a wave function, a function that assigns a complex number to each point ${\displaystyle x}$ at each time ${\displaystyle t}$. The parameter ${\displaystyle m}$ is the mass of the particle, and ${\displaystyle V(x,t)}$ is the potential energy function that represents the environment in which the particle exists. The constant ${\displaystyle i}$ is the imaginary unit, and ${\displaystyle \hbar }$ is the reduced Planck constant, which has units of action (energy multiplied by time).

Broadening beyond this simple case, the mathematical formulation of quantum mechanics developed by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl defines the state of a quantum mechanical system to be a vector ${\displaystyle |\psi \rangle }$ belonging to a separable complex Hilbert space ${\displaystyle {\mathcal {H}}}$. This vector is postulated to be normalized under the Hilbert space's inner product, that is, in Dirac notation it obeys ${\displaystyle \langle \psi |\psi \rangle =1}$. The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of square-integrable functions ${\displaystyle L^{2}}$, while the Hilbert space for the spin of a single proton is the two-dimensional complex vector space ${\displaystyle \mathbb {C} ^{2}}$ with the usual inner product.

Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are self-adjoint operators acting on the Hilbert space. A wave function can be an eigenvector of an observable, in which case it is called an eigenstate, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition. When an observable is measured, the result will be one of its eigenvalues with probability given by the Born rule: in the simplest case the eigenvalue ${\displaystyle \lambda }$ is non-degenerate and the probability is given by ${\displaystyle |\langle \lambda |\psi \rangle |^{2}}$, where ${\displaystyle |\lambda \rangle }$ is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ${\displaystyle \langle \psi |P_{\lambda }|\psi \rangle }$, where ${\displaystyle P_{\lambda }}$ is the projector onto its associated eigenspace.

A momentum eigenstate would be a perfectly monochromatic wave of infinite extent, which is not square-integrable. Likewise a position eigenstate would be a Dirac delta distribution, not square-integrable and technically not a function at all. Consequently, neither can belong to the particle's Hilbert space. Physicists sometimes regard these eigenstates, composed of elements outside the Hilbert space, as "generalized eigenvectors". These are used for calculational convenience and do not represent physical states. Thus, a position-space wave function ${\displaystyle \Psi (x,t)}$ as used above can be written as the inner product of a time-dependent state vector ${\displaystyle |\Psi (t)\rangle }$ with unphysical but convenient "position eigenstates" ${\displaystyle |x\rangle }$: $${\displaystyle \Psi (x,t)=\langle x|\Psi (t)\rangle .}$$