The Schrödinger wave equation is considered the most basic equation of non-relativistic quantum mechanics. In three spatial dimensions (that is, in $\mathbb{R}^3$ ) and for a single particle of mass $m$ , moving in a field of potential energy $V$ , the equation is $$ i \hbar\, \frac{\partial}{\partial t}\,\Psi(\vect r, t) = - \frac{\hbar^2}{2m}\cdot \triangle\, \Psi(\vect r, t) + V(\vect r, t)\, \Psi(\vect r, t), $$ where $\vect r := (x,y,z)$ is the position vector, $\hbar=h(2\pi)^{-1}$ , $h$ is Planck's constant, $\triangle$ denotes the Laplacian and $V(\vect r, t)$ is the value of the potential energy at point $\vect r$ and time $t$ . This equation is a second order homogeneous partial differential equation which is used to determine $\Psi$ , the so-called time-dependent wave function, a complex function which describes the state of a physical system at a certain point $\vect r$ and a time $t$ ($\Psi$ is thus a function of 4 variables: $x,y,z$ and $t$ ). The right hand side of the equation represents in fact the Hamiltonian operator (or energy operator) $H\Psi(\vect r, t)$ , which is represented here as the sum of the kinetic energy and potential energy operators. Informally, a wave function encodes all the information that can be known about a certain quantum mechanical system (such as a particle). The function's main interpretation is that of a position probability density for the particle ¹ (or system) it describes, that is, if $P(\vect r, t)$ is the probability that the particle is at position $\vect r$ at time $t,$ then an important postulate of M. Born states that $P(\vect r, t) = |\Psi(\vect r, t)|^2$ .

An example of a (relatively simple) solution of the equation is given by the wave function of an arbitrary (non-relativistic) free ² particle (described by a wave packet which is obtained by superposition of fixed momentum solutions of the equation). This wave function is given by: $$ \Psi(\vect r, t) = \int_{\mathcal{K}} A(\vect k) e^{i(\vect k\cdot \vect r - \hbar\vect{k}^2(2m)^{-1}\,t)}\,d\vect k $$ where $\vect k$ is the wave vector and $\mathcal{K}$ is the set of all values taken by $\vect k.$ For a free particle, the equation becomes $$ i \hbar\, \frac{\partial}{\partial t}\,\Psi(\vect r, t) = - \frac{\hbar^2}{2m}\cdot \triangle\, \Psi(\vect r, t $$ and it is easy to check that the aforementioned wave function is a solution.

An important special case is that when the energy $E$ of the system does not depend on time, i.e. $H\Psi = E\Psi$ , which gives rise to the time-independent Schrödinger equation: $$ E\Psi(\vect r) = - \frac{\hbar^2}{2m}\cdot \triangle\, \Psi(\vect r) + V(\vect r)\, \Psi(\vect r). $$

There are a number of generalizations of the Schrödinger equation, mostly in order to take into account special relativity, such as the Dirac equation (which describes a spin-$\frac{1}{2}$ particle with mass) or the Klein-Gordon equation (describing spin-$0$ particles).

Footnotes

... particle¹

This is in fact a little imprecise since the wave function is, in a way, a statistical tool: it describes a large number of identical and identically prepared systems. We speak of the wave function of one particle for convenience.

... free²

By free particle, we imply that the field of potential energy $V$ is everywhere $0.$