The Klein-Gordon Equation

The Klein-Gordon Equation

The previous derivation of the relationship between energy and momentum suggests that the rest mass frequency can be calculated as the norm of the energy-momentum vector, which of course we know already from Special Relativity.  Using the results from the previous post we can calculate:

which is a version of the Klein-Gordon equation.  This shows that the energy-momentum 4-vector is:

Putting the original rest mass frequency back into this equation gives:

Or, in terms of the particle rest mass:

                                   

                                   

                                   

The p⁰ term is the total energy of the particle and the p¹ to p³ components are the particle momenta along each axis times c.  (Note: the more conventional form of this vector is in units of momentum, with the energy component divided by c:

                                   

I prefer to keep this in energy units.)

Interpretation of the Klein-Gordon Equation

The Klein-Gordon equation is an energy equation.  The total energy  minus the kinetic energy  is the rest mass energy of the particle.

The Klein-Gordon equation is also a dispersion relationship between the wave function frequency, omega, and wave number, k.  The phase velocity is omega/k:

This phase velocity is infinite for a stationary particle, and goes to c for a very fast moving particle.  It does not appear that this phase velocity has any physical significance, however, so we will not consider it further.

The group velocity is the derivative of omega wrt k:

Or, alternatively

NOTE:  This last expression is only valid if there are no potentials.  Later we will extend this to particles in a potential field. 

If we plug in the expressions for omega and k in terms of the rest mass energy we get:

Next we will figure out how to use the Klein-Gordon equation to calculate the acceleration of a charged particle in an electric potential.  From that result we will then derive Schrodinger's Equation and show where the Lagrangian and principle of Least Action comes from.