Schrödinger’s wave equation

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 ${ℝ}^3$) and for a single particle of mass $m$, moving in a field of potential energy $V$, the equation is

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

where $𝒓:=(x,y,z)$ is the position vector, $\mathrm{\hslash }=h{(2\pi )}^{-1}$, $h$ is Planck’s constant, $\mathrm{△}$ denotes the Laplacian and $V(𝒓,t)$ is the value of the potential energy at point $𝒓$ 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 $𝒓$ 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) $H\mathrm{\Psi }(𝒓,t)$, 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 (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 postulate of M. Born states that $P(𝒓,t)={|\mathrm{\Psi }(𝒓,t)|}^2$.

An example of a (relatively simple) solution of the equation is given by the wave function of an arbitrary (non-relativistic) free particle (described by a wave packet which is obtained by superposition of fixed momentum solutions of the equation). This wave function is given by:

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

where $𝒌$ is the wave vector and $𝒦$ is the set of all values taken by $𝒌.$For a free particle, the equation becomes

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

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:

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

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).