Schrödinger’s wave equation


wave equationMathworldPlanetmath

The Schrödinger wave equation is considered to be the most basic equation of non-relativistic quantum mechanics. In three spatial dimensions (that is, in 3) and for a single particle of mass m, moving in a field of potential energy V, the equation is


where 𝒓:=(x,y,z) is the position vector, =h(2π)-1, h is Planck’s constant, denotes the LaplacianDlmfMathworld and V(𝒓,t) is the value of the potential energy at point 𝒓 and time t. This equation is a second orderPlanetmathPlanetmath homogeneousPlanetmathPlanetmathPlanetmathPlanetmath partial differential equation which is used to determine Ψ, the so-called time-dependent wave function, a complex function which describes the state of a physical system at a certain point 𝒓 and a time t (Ψ is thus a functionMathworldPlanetmath of 4 variables: x,y,z and t). The right hand side of the equation represents in fact the Hamiltonian operatorPlanetmathPlanetmath ( (or energy operator) HΨ(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 interpretationMathworldPlanetmathPlanetmath 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(𝒓,t) is the probability that the particle is at position 𝒓 at time t, then an important postulateMathworldPlanetmath of M. Born states that P(𝒓,t)=|Ψ(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 𝒌 is the wave vector and 𝒦 is the set of all values taken by 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. HΨ=EΨ, which gives rise to the time-independent Schrödinger equation:


There are a number of generalizationsPlanetmathPlanetmath of the Schrödinger equation, mostly in order to take into account special relativity, such as the Dirac equationMathworldPlanetmath (which describes a spin-12 particle with mass) or the Klein-Gordon equationMathworldPlanetmath (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