Chemistry - Klein-Gordon equation

The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is a relativistic version (describing scalar (or pseudoscalar) spinless particles) of the Schrödinger equation.

The Schrödinger equation for a free particle is

$\frac{\hat{\vec{p}}^2}{2m} \psi = i \frac{\partial}{\partial t}\psi$

where $\hat{\vec{p}} = -i\nabla$ is the momentum operator, using natural units where $\hbar=c=1$ .

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's special theory of relativity.

It is natural to try to use the identity from special relativity

$E = \sqrt{p^2 + m^2}$

for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

$\sqrt{(-i\nabla)^2 + m^2} \psi= i \frac{\partial}{\partial t}\psi$

This, however, is a cumbersome expression to work with because of the square root. Cumbersomeness, however, doesn't really count as an objection. But this equation, as it stands, is nonlocal.

Klein and Gordon instead worked with the square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads

$(\partial^2 + m^2) \psi = 0.$

The Klein-Gordon equation was actually first found by Schrödinger, before he made the discovery of the equation that now bears his name. He rejected it because he couldn't make it fit data (the equation doesn't take into account the spin of the electron); the way he found his equation was by making simplifications in the Klein-Gordon equation.

The Klein-Gordon equation may also be derived out of purely information-theoretic considerations. See extreme physical information.

In 1926, soon after the Schrödinger equation was introduced, Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein's method. Fock also determined the gauge theory for the wave equation. The Klein-Gordon equation for a free particle has a simple plane wave solution.

External links

Linear Klein-Gordon Equation at EqWorld: The World of Mathematical Equations.

Nonlinear Klein-Gordon Equation at EqWorld: The World of Mathematical Equations.

Categories: Partial differential equations | Equations | Special relativity | Waves