Klein-Gordon equation

The Klein-Gordon equation is a relativistic version (describing spinless particles) of the Schrödinger equation.

The Schrödinger's equation for a free particle is

<math>

\frac{\hat{\vec{p}}^2}{2m} \psi = i \frac{\partial}{\partial t}\psi </math> where <math>\hat{\vec{p}} = -i\nabla</math> is the momentum operator, using natural units where <math>\hbar=c=1</math>.

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 and use the identity from special relativity

<math>

E = \sqrt{p^2 + m^2} </math> for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

<math>

\sqrt{(-i\nabla)^2 + m^2} \psi= i \frac{\partial}{\partial t}\psi </math> This, however, is a cumbersome expression to work with because of the square root.

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

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(\partial^2 + m^2) \psi = 0. </math>

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.