A.14 The Klein-Gordon equation

The Schrödinger equation for the quantum wave function is based on the nonrelativistic expression for the energy of a particle. This addendum looks at the simplest relativistic equation for wave functions, called the Klein-Gordon equation. The discussion will largely be restricted to a spinless particle in empty space, where there is no potential energy. However, the Klein-Gordon equation is the first step to more complex relativistic equations.

Recall first how the Schrödinger equation arose. If there is no potential
energy, classical physics says that the energy

Then it applies the resulting operators on the wave function

Solutions with a definite value

Here

According to classical relativity however, the energy

The momentum can be identified with the same operator as before. But square roots of operators are very ugly. So the smart thing to do is to square both sides above. Making the same substitutions as for the Schrödinger equation and cleaning up then gives the “Klein-Gordon equation”

Solutions time-independent Klein-Gordon equation

or square Hamiltonian eigenvalue problem

This may be rewritten in a form so that both the Schrödinger equation and the Klein-Gordon equation are covered:

Here the constant

wave number.Note that the nonrelativistic energy does not include the rest mass energy. When that is taken into account, the Schrödinger expression for

Further note that relativistic or not, the magnitude of linear
momentum angular frequency

It may be noted that Schrödinger wrote down the Klein-Gordon equation first. But when he added the Coulomb potential, he was not able to get the energy levels of the hydrogen atom. To fix that problem, he wrote down the simpler nonrelativistic equation that bears his name. The problem in the relativistic case is that after you add the Coulomb potential to the energy, you can no longer square away the square root. Eventually, Dirac figured out how to get around that problem, chapter 12.12 and {D.81}. In brief, he assumed that the wave function for the electron is not a scalar, but a four-dimensional vector, (two spin states for the electron, plus two spin states for the associated antielectron, or positron.) Then he assumed that the square root takes a simple form for that vector.

Since this addendum assumes a particle in empty space, the problem with the Coulomb potential does not arise. But there are other issues. The good news is that according to the Klein-Gordon equation, effects do not propagate at speeds faster than the speed of light. That is known from the theory of partial differential equations. In classical physics, effects cannot propagate faster than the speed of light, so it is somewhat reassuring that the Klein-Gordon equation respects that.

Also, all inertial observers agree about the Klein-Gordon equation,
regardless of the motion of the observer. That is because all
inertial observers agree about the rest mass

But the bad news is that the Klein-Gordon equation does not
necessarily preserve the integral of the square magnitude of the wave
function. The Schrödinger equation implies that,
{D.32},

The wave function is then normalized so that the constant is 1. According to the Born statistical interpretation, chapter 3.1, the integral above represents the probability of finding the particle if you look at all possible positions. That must be 1 at whatever time you look; the particle must be somewhere. Because the Schrödinger equation ensures that the integral above stays 1, it ensures that the particle cannot just disappear, and that no second particle can show up out of nowhere.

But the Klein-Gordon equation does not preserve the integral above.
Therefore the number of particles is not necessarily preserved. That
is not as bad as it looks, anyway, because in relativity the
mass-energy equivalence allows particles to be created or destroyed,
chapter 1.1.2. But certainly, the interpretation of the
wave function is not a priori obvious. The integral that the
Klein-Gordon equation does preserve is, {D.32},

It is maybe somewhat comforting that according to this expression, the integral of

Another problem arises because even though the square energy

You might say, just ignore the negative energy possibility. But Dirac found that that does not work; you need both positive and negative energy states to explain such things as the hydrogen energy levels. The way Dirac solved the problem for electrons is to assume that all negative states are already filled with electrons. Unfortunately, that does not work for bosons, since any number of bosons can go into a state.

The modern view is to consider the negative energy solutions to
represent antiparticles. In that view, antiparticles have positive
energy, but move backwards in time. For example, Dirac’s negative
energy states are not electrons with negative energy, but positrons
with positive energy. Positrons are then electrons that move backward
in time. To illustrate the idea, consider two hypothetical wave
functions of the form

where

You see why so much quantum physics is done using nonrelativistic equations.