Solutions of the Klein-Gordon Equation

Non-plane-wave solutions of the Klein-Gordon equation using unconventional basic functions and coupling ansatz ... 

This important equation is known as the Klein-Gordon equation, and was proposed by various authors [6, 7, 8, 9] at much the same time. It is, however, an inconvenient equation to use, primarily because it involves a second-order differential operator with respect to time. Dirac therefore sought an equation linear in the momentum operator, whose solutions were also solutions of the Klein-Gordon equation. Dirac also required an equation which could more easily be generalised to take account of electromagnetic fields. The wave equation proposed by Dirac was [10]... 

It follows from (6) that every component of a solution of the free Dirac equation is a solution of the Klein-Gordon equation... 

Can we thus replace the free Dirac equation by Klein-Gordon equations, one for each component Certainly not, because the Dirac equation introduces a coupling between the components. A solution of the Dirac equation therefore contains information that is not contained in a spinor whose components are independent solutions of the Klein-Gordon equation. 

For a deeper understanding of the Klein-Gordon equation for a particle in field-free space we investigate its solutions. The Klein-Gordon eigenfunction in this case is given by the plane wave... 

Let us study the stability of the solution wq = Const found above. To do so, we perturb the Klein-Gordon equation around this solution. Here, we need only to consider the case 0 = 4>WQ=Const + where 6[Pg.142]

The time-dependent Schrodinger equation (2.43) presents a serious problem from the point of view of relativity theory it does not treat space and time in a symmetric way, because second-order derivatives of the wavefunction with respect to spatial coordinates are accompanied by a first-order derivative with respect to time. One way out, as actually proposed by Schrodinger and later known as the Klein-Gordon equation, would be to have also second-order derivatives with respect to time. However, that would lead to a total probability for the particle under consideration which would be a function of time, and to a variation of the number of particles of the universe (which, at the time, was completely unacceptable). In 1928, Dirac sought the solution for this problem, by accepting first-order derivation in the case of time and forcing the spatial derivatives to also be first order. This requires the wavefunction to have four components (functions of the spatial coordinates alone), often called a four-component spinor . 

The starting point it represents the Klein-Gordon equation of Section 2.2.3, Eq. (2.34) in the situation the wave-function solution is represented by the potential (its source) in a spherical (as a source) symmetry. 

Because of the unphysical feature of the Klein-Gordon density and the fact that spin does not emerge naturally (but would have to be included a posteriori as in the nonrelativistic framework) we are not able to deduce a fundamental relativistic quantum mechanical equation of motion for a freely moving electron. However, we may wonder which results of this section may be of importance for the derivation of such an equation of motion for the electron. Certainly, we would like to recover the plane wave solutions of Eq. (5.8) for the freely moving particle, but in order to introduce only a single integration constant (or the choice of a single initial value) for a positive definite density distribution we need to focus on first-order differential equations in time. These must also he first-order differential equations in space for the sake of Lorentz co-variance. 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

For the same reason it is not clear, how to modify the equation for the inclusion of external fields. The principle of minimal coupling p —> p — A, E E + V for the (scalar) square-root Klein-Gordon equation was critizised by J. Sucher [4], who states that there are solutions ip x) and electromagnetic potentials, such that the Lorentz transformed solution is not a solution of the equation with the Lorentz-transformed potentials. Moreover, the nonlocal nature of the equation means that the value of the potential at some point influences the wave function at other points and it is not clear at all how one can interpret this. 

The spin of elementary particles is not described by the square-root Klein-Gordon equation. The solutions of the square-root Klein-Gordon equation are scalar wave functions. Real electrons have spin and should be described by a matrix-wave equation. 

Schrodinger amplitude relation to Klein-Gordon amplitude, 500 Schrodinger equation, 439 adiabatic solutions, 414 as a unitary transformation, 481 for relativistic spin % particle, 538 for the component a, 410 in Fock representation, 459 in the q representation, 492 Schrodinger form of one-photon equation, 548... 

---