The Schr$\"{o}$dinger wave equation is considered to be 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

where $\boldsymbol{r}:=(x,y,z)$ is the position vector, $\hbar=h(2\pi)^{{-1}}$, $h$ is Planck’s constant, $\triangle$ denotes the Laplacian and $V(\boldsymbol{r},t)$ is the value of the potential energy at point $\boldsymbol{r}$ 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 $\boldsymbol{r}$ 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) $H\Psi(\boldsymbol{r},t)$, 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 particle¹¹This 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. (or system) it describes, that is, if $P(\boldsymbol{r},t)$ is the probability that the particle is at position $\boldsymbol{r}$ at time $t,$ then an important postulate of M. Born states that $P(\boldsymbol{r},t)=|\Psi(\boldsymbol{r},t)|^{2}$.

An example of a (relatively simple) solution of the equation is given by the wave function of an arbitrary (non-relativistic) free²²By free particle, we imply that the field of potential energy $V$ is everywhere $0.$ particle (described by a wave packet which is obtained by superposition of fixed momentum solutions of the equation). This wave function is given by:

where $\boldsymbol{k}$ is the wave vector and $\mathcal{K}$ is the set of all values taken by $\boldsymbol{k}.$ For a free particle, the equation becomes

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:

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).