# Schrödinger’s wave equation

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

 $i\hbar\,\frac{\partial}{\partial t}\,\Psi(\boldsymbol{r},t)=-\frac{\hbar^{2}}{% 2m}\cdot\triangle\,\Psi(\boldsymbol{r},t)+V(\boldsymbol{r},t)\,\Psi(% \boldsymbol{r},t),$

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 (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem) (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:

 $\Psi(\boldsymbol{r},t)=\int_{\mathcal{K}}A(\boldsymbol{k})e^{i(\boldsymbol{k}% \cdot\boldsymbol{r}-\hbar\boldsymbol{k}^{2}(2m)^{-1}\,t)}\,d\boldsymbol{k},$

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

 $i\hbar\,\frac{\partial}{\partial t}\,\Psi(\boldsymbol{r},t)=-\frac{\hbar^{2}}{% 2m}\cdot\triangle\,\Psi(\boldsymbol{r},t)$

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:

 $E\Psi(\boldsymbol{r})=-\frac{\hbar^{2}}{2m}\cdot\triangle\,\Psi(\boldsymbol{r}% )+V(\boldsymbol{r})\,\Psi(\boldsymbol{r}).$

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