Klein-gordon Equation Website Links For
Equation 		 

Information About

Klein-gordon Equation
  CATEGORIES ABOUT KLEIN-GORDON EQUATION
   
fundamental physics concepts
   
partial differential equations
   
equations
   
special relativity
   
waves
   
quantum field theory
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




DETAILS

The Schrödinger equation for a free particle is

:
rac{\mathbf{p}^2}{2m} \psi = i \hbar rac{\partial}{\partial t}\psi

where
:\mathbf{p} = -i \hbar \mathbf{
abla} is the momentum operator (
abla being the Del Operator ).

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein's Special Relativity .

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

:E = \sqrt{\mathbf{p}^2 c^2 + m^2 c^4}

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

: \sqrt{(-i\hbar\mathbf{
abla})^2 c^2 + m^2 c^4} \psi= i \hbar rac{\partial}{\partial t}\psi

This, however, is a cumbersome expression to work with because of the square root. In addition, this equation, as it stands, is Nonlocal .

Klein and Gordon instead worked with the more general ''square'' of this equation (the Klein-Gordon equation for a free particle), which in Covariant notation reads
:
(\Box^2 + \mu^2) \psi = 0.

where
: \mu = rac{mc}{\hbar} \, and
: \Box^2 = rac{1}{c^2} rac{\partial^2}{\partial t^2} -
abla^2\,. This operator is called the D'Alembert Operator .
Today this form is interpreted as the relativistic Field Equation for a scalar (i.e. Spin -0) particle.

The Klein-Gordon equation was allegedly 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 include the spin of the electron. The way Schrödinger found ''his'' equation was by making simplifications in the Klein-Gordon equation.

In 1926 , soon after the Schrödinger equation was introduced, Fock wrote an article about its generalization for the case of Magnetic Field s, where Force s 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.


RELATIVISTIC FREE PARTICLE SOLUTION


The Klein-Gordon equation for a free particle can be written as

:
\mathbf{
abla}^2\psi- rac{1}{c^2} rac{\partial^2}{\partial t^2}\psi
= rac{m^2c^2}{\hbar^2}\psi


with the same solution as in the non-relativistic case:

:
\psi(\mathbf{r}, t) = e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}


except with the constraint

:
-k^2+ rac{\omega^2}{c^2}= rac{m^2c^2}{\hbar^2}


Just as with the non-relativistic particle, we have for energy and momentum:

:
  Abla}\psi Angle \hbar\mathbf{k}
  \langle E Angle \langle \psi i\hbar rac{\partial}{\partial t}\psi angle = \hbar\omega
  \langle E Angle \langle \mathbf{p} angle c