Klein-Gordon and Schrodinger Equations

J. P. Vigier, Explicit mathematical construction of relativistic non-linear de Broglie waves described by three-dimensional (wave and electromagnetic) solitons piloted (controlled) by corresponding solutions of associated linear Klein-Gordon and Schrodinger equations, Found. Phys. 21(2) (1991). 

In going from the Schrodinger equation to the Klein-Gordon equation, we obtain the neeessary symmetry between spaee and time by having seeond-order derivatives throughout. It is usually written in a form that brings out its relativistic invarianee by using what is ealled/our-vector notation. We define a four-vector X to have components... 

The Schrodinger equation and the Klein-Gordon equation both involve second order partial derivatives, and to recover such an equation from the Dirac equation we can operate on equation 18.12 with the operator... 

The simplest derivation , given in many books, e.g. in chapter 4, was in fact similar to that used by Schrodinger to obtain an equation which falls short of the relativistic Schrodinger equation only by the absence of spin, a concept which had not yet arisen [1], This first quantum-mechanical wave equation is now known as the Klein-Gordon equation, and applies to particles without spin. 

The Klein-Gordon equation (Schrodinger s relativistic equation) has been used in the description of a relativistic particle with spin zero (see, e.g., Schiff, 1968) and can be treated using the so(2,1) algebraic methods (Barut, 1971 Cizek and Paldus, 1977, and references therein). It is obtained from the energy-momentum relationship... 

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 Schrodinger equation with this Hamiltonian is called the square-root Klein-Gordon equation because of its formal similarity to a square-root of the Klein-Gordon equation (Oscar Klein and Walter Gordon in fact had little to do with the square-root equation). Unfortunately, the meaning of the square-root Klein-Gordon equation is obscured by the following points. 

This equation has at least one advantage over the Schrodinger equation ct and x, y, z enter the equation on equal footing, whieh is required by special relativity. Moreover, the Fock-Klein-Gordon equation is invariant with respect to the Lorentz transformation, whereas the Schrodinger equation is not. This is a prerequisite of any relativity-consistent theory, and it is remarkable that such a simple derivation made the theory invariant. The invariance, however, does not mean that the equation is accurate. The Fock-Klein-Gordon equation describes a boson particle because vk is a usual scalar-type function, in contrast to what we will see shortly in the Dirac equation. 

If one uses the quantum postulates to introduce operators for E and p as it is done to derive the Schrodinger equation from energy conservation laws, one obtains an equation called the Klein-Gordon equation which involves second derivatives both in spatial coordinates and time. -This leads to the possibility that the probability density p = ij/ij/ could be negative which makes it difficult to interpret p. 

Now, in a wave equation analogous to the Schrodinger equation, the time-like variable E/c and the space-like momenta p should appear raised to the same power, since space and time have to be treated on the same footing. Obviously, the expression (3) as it stands leads to an equation which is of second order in both E/c and p, a relativistic wave equation known as the Klein-Gordon equation, valid for particles with spin 0. A linearization of the equation, i.e., a factorization according to... 

What to insert as the operator H of the energy El This was done by Schrodinger (even before Fock, Klein and Gordon). Schrodinger inserted what he had on the right-hand side of his time-dependent equation... 

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