To be used in future
 Section 6.1: Probability currents Up Chapter 6: Relativistic wave equations Section 6.3: Dirac equation 

6.2 Klein-Gordon equation↓

6.2.1 relativistic energy momentum relation↓

We shall now attempt to find candidates for a relativistic wave equation. One natural way is to start from the correspondence principle↓. We know that the relativistic energy (it is probably better not to consider Hamiltonians at this stage) for a particle of (rest-)mass and momentum is If we interpret this as an energy operator↓, we run into the difficult problem of having to calculate the square-root of a differential operator, which seems rather unattractive!

6.2.2 Klein-Gordon wave equation↓

One obvious solution is to square the expression on both sides before using the correspondence principle; this leads to the rather symbolic wave equation Using the fact that the energy operator is , as we know from non-relativistic quantum mechanics, we find the Klein Gordon equation This is essentially the equation for light waves, with one additional term, the factor that includes the mass of the particle.

6.2.3 Some simple solutions↓

The Klein-Gordon equation is often written in the compact form with the “d’Alembertian”
Here we use the notation We raise the index by with We also use the “Einstein summation convention”: unless explicitly indicated, any repeated index (one that appear once as a sub- and once as a superscript) is summed over.
We tackle the solution of this equation by separating variables in the standard form and find that so this plane wave (a flat wavefront wave propagating with momentum ) is a solution to the Klein-Gordon equation as long as we require that Note the fact that we can have negative energy, a surprising and to some people worrying consequence of this approach!

6.2.4 Lorentz invariance and external potentials↓↓↓

If we look carefully, we can see that Eq. (↓) is written in a natural Lorentz invariant form, since does not change when we change frames by a Lorentz transformation (please prove this yourself, if it is not immediately obvious from the form (↓)). If we couple an electromagnetic field to the equation, the logical way to do it is to replace We speak of a “vector potential” in this case—i.e., this potential transforms as a vector under Lorentz transformations. [Z]  [Z] Usually written as with , see Ref. [13] The alternative is to add a “scalar potential” , one that is invariant under a Lorentz transformation. This means adding the potential to , the scalar part in the equation--since the rest-mass is Lorentz invariant. This can also be interporetted to day that the mass of our particle depends on its position in space... . Thus the most general form of the Klein-Gordon equation in an external field is

6.2.5 Probability and currents↓↓

If we try and find a continuity equation for the Klein-Gordon equation, which in relativity must be of the form we see that we can take over as almost unchanged from Eq. (↓). The density differs from Eq. (↓) by a time derivative, since the Klein-Gordon equation is second order in time, The structure is quite attractive, The continuity equation can now be written where the factor in the time derivative is ubiquitous in relativistic problems. From Eq. (↓) we find that for a plane wave of the form we have

6.2.6 Issues

Clearly having solutions with negative energy is an important issue! This leads to the fact that the probability density can be negative as well as positive, depending on the sign of , see Eq. (↓). That really means that as it stands can not be a probability density. Since it satisfies a continuity condition, it must be quite close! One normally argues that the negative energy solutions are linked to antiparticles, but we shall not pursue this here in any detail. Suffice it to say that this both resolves the problems caused by negative energy solutions and negative .↓
 Section 6.1: Probability currents Up Chapter 6: Relativistic wave equations Section 6.3: Dirac equation