What is the Klein-Gordon equation

[Greg Bernhardt]

Definition/Summary

In this library item, some properties and interpretations of the Klein-Gordon equation (KG) will be covered. We will first focus on its usage in Relativistic Quantum Mechanics (RQM) and then examine it in Quantum Field Theory (QFT).

The Klein-Gordon equation is:

$$\frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0,$$

this is for a free particle, but what $\psi = \psi (t\, , \vec{x})$ is depends on the interpretation as we will see further on.

This free KG can be formulated in a covariant fashion:
$$\left( (i \partial_\mu)(i \partial ^\mu) -m^2 \right) \psi = 0$$

where we have used units where c = 1 and $\hbar$ = 1.

Or, by further simplified notation, introducing the d'Alembert-operator:
$$(\Box^2 + m^2) \psi = 0 .$$

For a charged particle, charge e, subject to a electromagnetic potential, we substitute:
$$i\partial _\mu \rightarrow i\partial _\mu - eA_\mu(x),$$

where $A_\mu (x)= (\phi (x), \vec{A}(x) ).$

Equations

$$\frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0,$$

Extended explanation

Introduction
The Schrödinger equation in Quantum Mechanics is mathematically a first order differential equation. The Schrödinger equation describes the dynamical evolution of a systems wave function. In order to fulfil the basic requirement of special relativity, that nothing can travel faster than the light speed in vacuum, one must consider another equation than the Schrödinger equation.

The Schrödinger equation is built on the non relativistic formulation of mechanics, a classical Hamiltonian with kinetic energy: $p^2/2m$ which in first quantization reads: $-(\hbar^2/ 2m )\nabla^2$. The energy of the wave function is then given by the operator $i\hbar\frac{\partial}{\partial t}$ since the Hamiltonian is the generator of time-evolution.

Thus a transition from a place in space to another, given by:
$$<x_1|U (t) |x_0> = <x_1|e^{i(p^2/2m)t} = ... = (m/2\pi t\, i)^{3/2}e^{i(x_1-x_0)^2 m/2t}$$
which is non-zero for all t's! Which is in violation with causality - only those points which are within time = (x1-x0)/c should be able to reach (in a fancier formulation; the transition amplitude should vanish outside the light-cone).

Thus one tried another equation for relativistic quantum physics.
(in reality, it was the other way around, one tried to formulate relativistic quantum physics before the Schrödinger equation. But one encountered another problem, which we will see in the next section.)

Relativistic Equation
The simplest and most straightforward way to do this is the use the relativistic formulation of energy:
$$E^2 = p^2 + m^2$$
and using the same first-quantization procedure as in the non-relativistic case, the result is the Klein-Gordon equation, which is second order differential equation:

$$\frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0,$$

Which can be formulated in the covariant fashion:
$$\left( (i \partial_\mu)(i \partial ^\mu) -m^2 \right) \psi = 0$$

Or, by further simplified notation, introducing the d'Alembert-operator:
$$(\Box^2 + m^2) \psi = 0 .$$

$$( \Box^2 = \frac{\partial^2}{\partial t^2} - \nabla^2 )$$

The inclusion of a an electromagnetic potential is straightforward, by using the substitution:
$$i\partial _\mu \rightarrow i\partial _\mu - eA_\mu(x),$$
where $A_\mu (x)= (\phi (x), \vec{A}(x) )$, is the electromagnetic four-potential.

Plane-wave solution
The plane-wave solutions for the free KG are:
$$f_p^{(\pm)} (x) = Ne^{\mp i p\,x}$$

with normalization:

$$N = (2E_p)^{-1/2},$$

$$E_p = \sqrt{\vec{p}^2 + m^2} > 0 .$$

These plane waves fulfil (equal time) orthogonality relations:

$$i\int d\vec{x} \, ( f_{q}^{(\pm)*}(\partial_t f_{p}^{(\pm)}) - f_{p}^{(\pm)}(\partial_t f_{q}^{(\pm)*})) = \pm \delta (\vec{q}-\vec{p})$$

$$i\int d\vec{x} \, ( f_{q}^{(\pm)*}(\partial_t f_{p}^{(\mp)}) - f_{p}^{(\mp)}(\partial_t f_{q}^{(\pm)*})) = 0$$

Quantum properties
The KG equation will fulfil causality, but it will not fulfil the requirement of quantum mechanics. In quantum mechanics, the function is a wave function which resembles the probability density amplitude. Thus, the norm should be positive (negative probability is what?). A simple, but for this library item, a quite length derivation of the associated current and density of the KG equation will show that the density can be NEGATIVE! (this is a consequence of having an equation of second order in derivatives). The density is negative for the plane-wave solutions:

$$f_p^{(-)} (x) = Ne^{+ i p\,x}$$

which we call the negative energy/frequency solution (the other solution is called positive energy solution).

Thus, we need to determine what we want to describe with the function that solves the KG equation. One way is actually to forget about quantum mechanics, and try to formulate a new framework - Quantum Fields (which will be discussed later in this item and also in the library item on Quantum Field Theory).

Another way, is to interpret the density not as probability, but as charge density (in the case for electrically charged particles). Thus we have now, in a way, postulated the existence of antiparticles - with same mass but opposite charge (more on this further down in this post and also in the library item on antiparticles).

Scattering (RQM)
Let us now examine a bit what we can do with the second suggestion above, namely that we have a relativistic quantum mechanics (RQM) framework, and let the positive energy solutions describe the probability density amplitude of a particle with charge e and the negative energy solution be the same but for particle with charge -e.

In particle physics, we want to calculate the probability that particles of different kinds in an interaction produces a certain final state. As an example, the probability that an electron with incoming momentum p_1 interacts with a nucleus and is detected later with momentum p_2.

More..

Then there will be, after creation of a general post about QFT, canonical quantization and Lagrangian theory of KG, then some more features of KG in QFT.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[Wrichik Basu]

Quite interesting introduction to the KG equations for beginners. Shouldn't these be moved to insights?