The Klein-Gordon Equation

1This equation is named after Oskar Klein and Walter Gordon, who in 1927 forwarded it as a solution for a relativistic electron. However, the subsequent development of the Dirac equation, in 1928, was shown to describe the spinning electron, while the Klein–Gordon equation describes a ‘particle’ with mass, but no spin, e.g. a pion. The Klein–Gordon equation is often introduced as a relativistic ‘derivative’ of the Schrödinger equation because it is rooted in the same definition of the quantized wave operators associated with some wave function [Ψ]. However, as will be shown, the format of the Klein-Gordon equation, like the Dirac equation, is essentially a revised form of the relativistic energy equation. This said, the Klein-Gordon equation predates the Dirac equation and therefore may be said to represent the first step toward a description that is now often tagged as Relativistic Quantum Mechanics (RQM) . Subsequent discussions will continue the development of quantum physics in terms of a number of steps along the following path:

  • Non-Relativistic Quantum Mechanics
  • Relativistic Quantum Mechanics  ←You are here
  • Quantum Field Theory

As such, the Klein-Gordon equation is an important step from NRQM to RQM. However, before proceeding to discuss this equation, it might be useful to review some aspects of Schrodinger’s wave mechanics, which are to be merged with special relativity. In the context of non-relativistic-QM (NRQM), Schrodinger’s equation was thought to provide a fundamental description of nature in terms of the 'quantized wave operators' of energy [E] and momentum [p]:

[1]      1

We can highlight the role of these operators in Schrodinger’s equation, while noting the fact that these operators still reflect the units of energy and momentum, such that the wave function [Ψ] they operate on must have no units. Based on [1], we can reverse–engineer the time-dependent Schrodinger equation for a free particle, i.e. subject to no potential field [U=0], as follows:

[2]      2

The side note attached to the last equation is simply to highlight that the energy in this case is defined in terms of its kinetic energy only, as we have assume [U=0]. This issue will be highlighted again at the end of this discussion in a wider context. However, at this point, we might still question what [2] is really telling us unless we can quantify the form of the wave function [Ψ]. For simplicity, we will start by describing [Ψ] in terms of some basic formulation of a 1-dimensional wave equation of the type shown in [3]:

[3]      3

Before outlining the suitability of [3] to describe a matter wave, it might be worth highlighting that [3] can be modified to reflect both quantum operators in [1] – see [4]. In this respect, it can be said that the mechanics of a wave allow both energy and momentum to be transported in space and time without any obvious reference to the semantics of a particle, e.g. its mass. This point is raised simply because the following discussion will indirectly touch on the wave-particle duality issue and include the semantics of mass [m] within many of the equations. Whether this reflects a true description of the fundamental nature of the quantum realm is left as an open question.

[4]      4

However, [3] reflects a generalised waveform associated with a continuous wave that is conceptually infinite in space, whereas we need a wave function [Ψ] that will be more representative of a matter wave that is localised in space. Again, at this stage, this issue is simply being highlighted, although it will be discussed further when we consider the ‘implications of the quantum wave function that results from deBroglie hypothesis. As implied, the Schrodinger equation in [2] does not account for the effects of special relativity on both energy [E] and momentum [p], which is often linked to its effects on the mass of a particle. So, with these contextual preliminaries in place, we might now consider the derivation of the Klein-Gordon equation, which can be shown to be rooted in the relativistic energy equation shown in [5]:

[5]      5

Based on [1], we might substitute for both energy and momentum in terms of the quantized operators linked to some wave function [Ψ]:

[6]      6

In some respect, [6] already contains all the essential features of the Klein-Gordon equation, although we might need to re-arrange some of the terms so that the format becomes more recognisable to that normally adopted in the standard form:

[7]      7

However, from  the perspective of trying to understand what is going on here, it might be useful to reflect a little further on the implications of [6] and [7]. To some extent, we are trying to examine the duality of the wave-particle descriptions embedded in both the Schrodinger and Klein-Gordon equations. For example, if we do a 2nd order differentiation of the wave equation in [3] we get:

[8]      8

As such, [8] allows us to examine the equality of [5] and [7] by replacing the differential terms with the wave number [κ] and the angular frequency [ω], which returns [7] to [5]:

[9]      9

However, we might also want to examine [7] in terms of the classical wave equation of motion that has the following general form:

[10]    10

In terms of a classical wave, we might interpret [10] by saying that the 2nd order derivative, i.e. acceleration of some amplitude [A], with respect to time [t] and space [x] is directly proportional to the square of the propagation velocity [v]. However, if we re-arrange [7] to provide a more direct comparison to [10], we get the form shown in [11]:

[11]    11

At first glance, the transitional formats in [11] may not appear to be that helpful in the examining of the wave-particle duality description. However, the following equations highlight that each term in the last equation in [11] is an expression of wave frequency [f]:

[12]    12

Therefore, we might characterise the form of [11] as follows:

[13]    13

Here [f1] can be associated with the relativistic momentum [p], while [f2] is associated with the rest mass [m0], which both appear in [5]. However, the inference of equating the Planck and Einstein energy equations, as per [12], suggests that [13] is simply a reflection of the energy components being considered:

[14]    14

In [14], we can now see more clearly that the scope of the energy being considered in both the Schrodinger equation in [2] and the Klein-Gordon equation in [7]. As such, it might be highlighted that in non-relativistic cases, the energy associated with the rest mass would swamp the kinetic energy and so on examination of [2], we see that only the kinetic energy is taken into consideration. In contrast, the Klein-Gordon equation in [7] assumes that the relativistic kinetic energy [pc] is comparable to the rest energy [m0c2] and so both terms are taken into account. It has also to be highlighted that we have chosen to simplify this discussion by assuming the quantum particles in question are free from any effects of a potential field, which we might linked to the four fundamental forces of nature. If you pursue this line of thought, it would seem that the quantum description has the ‘potential’ for much complexity, which has not really been taken into account, so far.