PlanetMath: Schrödinger's wave equation

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Schrödinger's wave equation

(Definition)

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 , moving in a field of potential energy , the equation is

$\displaystyle i \hbar\, \frac{\partial}{\partial t}\,\Psi(\boldsymbol{r}, t) = ... ...le\, \Psi(\boldsymbol{r}, t) + V(\boldsymbol{r}, t)\, \Psi(\boldsymbol{r}, t),$

where $\boldsymbol{r} := (x,y,z)$ is the position vector, $\hbar=h(2\pi)^{-1}$ ,

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

. 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

( $\Psi$ is thus a function of 4 variables:

and

). The right hand side of the equation represents in fact the Hamiltonian (or energy operator) $H\Psi(\boldsymbol{r}, t)$ , which is represented here as the sum of the kintetic 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 $P(\boldsymbol{r}, t)$ is the probability that the particle is at position $\boldsymbol{r}$ at time

then an important postulate of M. Born states that $P(\boldsymbol{r}, t) = \vert\Psi(\boldsymbol{r}, t)\vert^2$ .

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:

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

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

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

and it is easy to check that the aforementioned wave function is a solution.

An important special case is that when the energy of the system does not depend on time, i.e. $H\Psi = E\Psi$ , which gives rise to the time dependant Schrödinger equation:

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

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).

Footnotes

... particle 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.
... free ²: By free particle, we imply that the field of potential energy is everywhere

"Schrödinger's wave equation" is owned by Cosmin. [ full author list (2) | owner history (2) ]

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Other names:

Schrödinger's equation, time-dependent Schrödinger wave equation

Also defines:

wave function

Keywords:

quantum mechanics, wave equation, partial differential equation

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Cross-references: Klein-Gordon equation, Dirac equation, vector, imply, solution, induces, postulate, interpretation, operator, right hand side, variables, complex function, partial differential equation, homogeneous, second order, point, Laplacian, position vector, dimensions, equation
There are 5 references to this entry.

This is version 16 of Schrödinger's wave equation, born on 2005-02-16, modified 2007-06-25.
Object id is 6756, canonical name is SchrodingersWaveEquation.
Accessed 7263 times total.

Classification:

AMS MSC:	35Q40 (Partial differential equations :: Equations of mathematical physics and other areas of application :: Equations from quantum mechanics)
	81Q05 (Quantum theory :: General mathematical topics and methods in quantum theory :: Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other quantum-mechanical equations)

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