Schrödinger’s wave equation
The Schrdinger wave equation is considered to be the most basic equation of non-relativistic quantum mechanics. In three spatial dimensions (that is, in ) and for a single particle of mass , moving in a field of potential energy , the equation is
where is the position vector, , is Planck’s constant, denotes the Laplacian and is the value of the potential energy at point and time . This equation is a second order homogeneous partial differential equation which is used to determine , the so-called time-dependent wave function, a complex function which describes the state of a physical system at a certain point and a time ( is thus a function of 4 variables: and ). The right hand side of the equation represents in fact the Hamiltonian operator (http://planetmath.org/HamiltonianOperatorOfAQuantumSystem) (or energy operator) , 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 is the probability that the particle is at position at time then an important postulate of M. Born states that .
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 is everywhere 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 is the wave vector and is the set of all values taken by 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 of the system does not depend on time, i.e. , 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- particle with mass) or the Klein-Gordon equation (describing spin- particles).
|Title||Schrödinger’s wave equation|
|Date of creation||2013-03-22 15:02:31|
|Last modified on||2013-03-22 15:02:31|
|Last modified by||Cosmin (8605)|
|Synonym||time-independent Schrödinger wave equation|