# Schrödinger’s wave equation

The *Schr$\xf6$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

$$i\mathrm{\hslash}\frac{\partial}{\partial t}\mathrm{\Psi}(\bm{r},t)=-\frac{{\mathrm{\hslash}}^{2}}{2m}\cdot \mathrm{\u25b3}\mathrm{\Psi}(\bm{r},t)+V(\bm{r},t)\mathrm{\Psi}(\bm{r},t),$$ |

where $\bm{r}:=(x,y,z)$ is the position vector, $\mathrm{\hslash}=h{(2\pi )}^{-1}$, $h$ is Planck’s constant, $\mathrm{\u25b3}$ denotes the Laplacian^{} and $V(\bm{r},t)$ is the value of the potential energy at point $\bm{r}$ and time $t$.
This equation is a second order^{} homogeneous^{} partial differential equation which is used to determine $\mathrm{\Psi}$, the so-called *time-dependent wave function*, a complex function which describes the state of a physical system at a certain point $\bm{r}$ and a time $t$ ($\mathrm{\Psi}$ 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^{} (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem) (or energy operator) $H\mathrm{\Psi}(\bm{r},t)$, 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}^{1}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. (or system) it describes, that is, if $P(\bm{r},t)$ is the probability that the particle is at position $\bm{r}$ at time $t,$ then an important postulate^{} of M. Born states that $P(\bm{r},t)={|\mathrm{\Psi}(\bm{r},t)|}^{2}$.

An example of a (relatively simple) solution of the equation is given by the wave function of an arbitrary (non-relativistic) free^{2}^{2}By 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:

$$\mathrm{\Psi}(\bm{r},t)={\int}_{\mathcal{K}}A(\bm{k}){e}^{i(\bm{k}\cdot \bm{r}-\mathrm{\hslash}{\bm{k}}^{2}{(2m)}^{-1}t)}\mathit{d}\bm{k},$$ |

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

$$i\mathrm{\hslash}\frac{\partial}{\partial t}\mathrm{\Psi}(\bm{r},t)=-\frac{{\mathrm{\hslash}}^{2}}{2m}\cdot \mathrm{\u25b3}\mathrm{\Psi}(\bm{r},t)$$ |

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. $H\mathrm{\Psi}=E\mathrm{\Psi}$, which gives rise to the *time-independent Schrödinger equation*:

$$E\mathrm{\Psi}(\bm{r})=-\frac{{\mathrm{\hslash}}^{2}}{2m}\cdot \mathrm{\u25b3}\mathrm{\Psi}(\bm{r})+V(\bm{r})\mathrm{\Psi}(\bm{r}).$$ |

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).

^{}Title | Schrödinger’s wave equation |

Canonical name | SchrodingersWaveEquation |

Date of creation | 2013-03-22 15:02:31 |

Last modified on | 2013-03-22 15:02:31 |

Owner | Cosmin (8605) |

Last modified by | Cosmin (8605) |

Numerical id | 28 |

Author | Cosmin (8605) |

Entry type | Topic |

Classification | msc 81Q05 |

Classification | msc 35Q40 |

Synonym | Schrödinger’s equation |

Synonym | time-independent Schrödinger wave equation |

Related topic | SchrodingerOperator |

Related topic | HamiltonianOperatorOfAQuantumSystem |

Related topic | Quantization |

Related topic | DiracEquation |

Related topic | KleinGordonEquation |

Related topic | PauliMatrices |

Related topic | DAlembertAndDBernoulliSolutionsOfWaveEquation |

Defines | wave function |